Chapter 9: Trigonometry | Turan Arıcan

CHAPTER 9 Trigonometry

Riverpole by Vaughn Warren – Kamloops, BC.

Trigonometry is a part of geometry that takes its origin in the ancient study of the relationship of the sides and angles of a right triangle. “Trigon” from Greek means triangle and “metron” means measure.

Applications of trigonometry are essential to many disciplines like carpentry, engineering, surveying, and astronomy, just to name a few.

How tall is the Riverpole? Do we have to climb the pole to find out? Fortunately, with the knowledge of trigonometry, we can find out the measurements of tall objects without too much hassle.

In this chapter we will explore the basic properties of angles and triangles, and the applications of the Pythagorean Theorem and trigonometric ratios.

9.1 Use Properties of Angles, Triangles, and the Pythagorean Theorem

Learning Objectives

By the end of this section, you will be able to:

Use the properties of angles
Use the properties of triangles
Use the Pythagorean Theorem

Use the Properties of Angles

Are you familiar with the phrase ‘do a $180'?$ It means to make a full turn so that you face the opposite direction. It comes from the fact that the measure of an angle that makes a straight line is $180$ degrees. See (Figure 1).

Figure 1

An angle is formed by two rays that share a common endpoint. Each ray is called a side of the angle and the common endpoint is called the vertex. An angle is named by its vertex. In (Figure 2), $mathematical expression$ is the angle with vertex at point $A$ . The measure of $mathematical expression$ is written $mathematical expression$ .

$mathematical expression$ is the angle with vertex at $mathematical expression$ .

The image is an angle made up of two rays. The angle is labeled with letter A.

Figure 2

We measure angles in degrees, and use the symbol ° to represent degrees. We use the abbreviation $m$ to for the measure of an angle. So if $mathematical expression$ is 27°, we would write $mathematical expression$ .

If the sum of the measures of two angles is $180$ °, then they are called supplementary angles. In (Figure 3), each pair of angles is supplementary because their measures add to $180$ °. Each angle is the supplement of the other.

The sum of the measures of supplementary angles is $180$ °.

Part a shows a 120 degree angle next to a 60 degree angle. Together, the angles form a straight line. Below the image, it reads 120 degrees plus 60 degrees equals 180 degrees. Part b shows a 45 degree angle attached to a 135 degree angle. Together, the angles form a straight line. Below the image, it reads 45 degrees plus 135 degrees equals 180 degrees.

Figure 3

If the sum of the measures of two angles is $90$ °, then the angles are complementary angles. In (Figure 4), each pair of angles is complementary, because their measures add to $90$ °. Each angle is the complement of the other.

The sum of the measures of complementary angles is $90$ °.

Part a shows a 50 degree angle next to a 40 degree angle. Together, the angles form a right angle. Below the image, it reads 50 degrees plus 40 degrees equals 90 degrees. Part b shows a 60 degree angle attached to a 30 degree angle. Together, the angles form a right angle. Below the image, it reads 60 degrees plus 30 degrees equals 90 degrees.

Figure 4

Supplementary and Complementary Angles

If the sum of the measures of two angles is $180$ °, then the angles are supplementary.

If $mathematical expression$ and $mathematical expression$ are supplementary, then $mathematical expression$ °.

If the sum of the measures of two angles is $90$ °, then the angles are complementary.

If $mathematical expression$ and $mathematical expression$ are complementary, then $mathematical expression$ °.

In this section and the next, you will be introduced to some common geometry formulas. We will adapt our Problem Solving Strategy for Geometry Applications. The geometry formula will name the variables and give us the equation to solve.

In addition, since these applications will all involve geometric shapes, it will be helpful to draw a figure and then label it with the information from the problem. We will include this step in the Problem Solving Strategy for Geometry Applications.

HOW TO: Use a Problem Solving Strategy for Geometry Applications

Read the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information.
Identify what you are looking for.
Name what you are looking for and choose a variable to represent it.
Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
Solve the equation using good algebra techniques.
Check the answer in the problem and make sure it makes sense.
Answer the question with a complete sentence.

The next example will show how you can use the Problem Solving Strategy for Geometry Applications to answer questions about supplementary and complementary angles.

EXAMPLE 1

An angle measures $40$ °. Find a) its supplement, and b) its complement.

Solution

a)
Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.
Step 3. Name. Choose a variable to represent it.
Step 4. Translate. Write the appropriate formula for the situation and substitute in the given information.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.

b)
Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.
Step 3. Name. Choose a variable to represent it.
Step 4. Translate. Write the appropriate formula for the situation and substitute in the given information.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.

TRY IT 1.1

An angle measures $25$ °. Find its: a) supplement b) complement.

Show answer

155°
65°

TRY IT 1.2

An angle measures $77$ °. Find its: a) supplement b) complement.

Show answer

103°
13°

Did you notice that the words complementary and supplementary are in alphabetical order just like $90$ and $180$ are in numerical order?

EXAMPLE 2

Two angles are supplementary. The larger angle is $30$ ° more than the smaller angle. Find the measure of both angles.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.
Step 3. Name. Choose a variable to represent it. The larger angle is 30° more than the smaller angle.
Step 4. Translate. Write the appropriate formula and substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.

TRY IT 2.1

Two angles are supplementary. The larger angle is $100$ ° more than the smaller angle. Find the measures of both angles.

Show answer

40°, 140°

TRY IT 2.2

Two angles are complementary. The larger angle is $40$ ° more than the smaller angle. Find the measures of both angles.

Show answer

25°, 65°

Use the Properties of Triangles

What do you already know about triangles? Triangle have three sides and three angles. Triangles are named by their vertices. The triangle in (Figure 5) is called $mathematical expression$ , read ‘triangle $ABC$ ’. We label each side with a lower case letter to match the upper case letter of the opposite vertex.

$mathematical expression$ has vertices $mathematical expression$ and sides $mathematical expression$

The vertices of the triangle on the left are labeled A, B, and C. The sides are labeled a, b, and c.

Figure 5

The three angles of a triangle are related in a special way. The sum of their measures is $180$ °.

$mathematical expression$ °

Sum of the Measures of the Angles of a Triangle

For any $mathematical expression$ , the sum of the measures of the angles is $180$ °.

$mathematical expression$ °

EXAMPLE 3

The measures of two angles of a triangle are $55$ ° and $82$ °. Find the measure of the third angle.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.
Step 3. Name. Choose a variable to represent it.
Step 4. Translate. Write the appropriate formula and substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.

TRY IT 3.1

The measures of two angles of a triangle are $31$ ° and $128$ °. Find the measure of the third angle.

Show answer

21°

TRY IT 3.2

A triangle has angles of $49$ ° and $75$ °. Find the measure of the third angle.

Show answer

56°

Right Triangles

Some triangles have special names. We will look first at the right triangle. A right triangle has one $90$ ° angle, which is often marked with the symbol shown in (Figure 6).

A right triangle is shown. The right angle is marked with a box and labeled 90 degrees.

Figure 6

If we know that a triangle is a right triangle, we know that one angle measures $90$ ° so we only need the measure of one of the other angles in order to determine the measure of the third angle.

EXAMPLE 4

One angle of a right triangle measures $28$ °. What is the measure of the third angle?

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.
Step 3. Name. Choose a variable to represent it.
Step 4. Translate. Write the appropriate formula and substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.

TRY IT 4.1

One angle of a right triangle measures $56$ °. What is the measure of the other angle?

Show answer

34°

TRY IT 4.2

One angle of a right triangle measures $45$ °. What is the measure of the other angle?

Show answer

45°

In the examples so far, we could draw a figure and label it directly after reading the problem. In the next example, we will have to define one angle in terms of another. So we will wait to draw the figure until we write expressions for all the angles we are looking for.

EXAMPLE 5

The measure of one angle of a right triangle is $20$ ° more than the measure of the smallest angle. Find the measures of all three angles.

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the measures of all three angles
Step 3. Name. Choose a variable to represent it. Now draw the figure and label it with the given information.
Step 4. Translate. Write the appropriate formula and substitute into the formula.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.

TRY IT 5.1

The measure of one angle of a right triangle is $50$ ° more than the measure of the smallest angle. Find the measures of all three angles.

Show answer

20°, 70°, 90°

TRY IT 5.2

The measure of one angle of a right triangle is $30$ ° more than the measure of the smallest angle. Find the measures of all three angles.

Show answer

30°, 60°, 90°

Similar Triangles

When we use a map to plan a trip, a sketch to build a bookcase, or a pattern to sew a dress, we are working with similar figures. In geometry, if two figures have exactly the same shape but different sizes, we say they are similar figures. One is a scale model of the other. The corresponding sides of the two figures have the same ratio, and all their corresponding angles are have the same measures.

The two triangles in (Figure 7) are similar. Each side of $mathematical expression$ is four times the length of the corresponding side of $mathematical expression$ and their corresponding angles have equal measures.

$mathematical expression$ and $mathematical expression$ are similar triangles. Their corresponding sides have the same ratio and the corresponding angles have the same measure.

Two triangles are shown. They appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled 16, the side across from B is labeled 20, and the side across from C is labeled 12. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 4, the side across from Y is labeled 5, and the side across from Z is labeled 3. Beside the triangles, it says that the measure of angle A equals the measure of angle X, the measure of angle B equals the measure of angle Y, and the measure of angle C equals the measure of angle Z. Below this is the proportion 16 over 4 equals 20 over 5 equals 12 over 3.

Figure 7

Properties of Similar Triangles

If two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths are in the same ratio.

The length of a side of a triangle may be referred to by its endpoints, two vertices of the triangle. For example, in $mathematical expression$

$mathematical expression$

We will often use this notation when we solve similar triangles because it will help us match up the corresponding side lengths.

EXAMPLE 6

$mathematical expression$ and $mathematical expression$ are similar triangles. The lengths of two sides of each triangle are shown. Find the lengths of the third side of each triangle.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.	The figure is provided.
Step 2. Identify what you are looking for.	The length of the sides of similar triangles
Step 3. Name. Choose a variable to represent it.	Let a = length of the third side of $mathematical expression$ y = length of the third side $mathematical expression$
Step 4. Translate.	The triangles are similar, so the corresponding sides are in the same ratio. So $AB over XY=BC over YZ=AC over XZ$ Since the side $AB=4$ corresponds to the side $XY=3$ , we will use the ratio $AB over XY=4 over 3$ to find the other sides. Be careful to match up corresponding sides correctly.
Step 5. Solve the equation.
Step 6. Check.
Step 7. Answer the question.	The third side of $mathematical expression$ is 6 and the third side of $mathematical expression$ is 2.4.

TRY IT 6.1

$mathematical expression$ is similar to $mathematical expression$ . Find $a$ .

Show answer

TRY IT 6.2

$mathematical expression$ is similar to $mathematical expression$ . Find $y$ .

Show answer

22.5

Use the Pythagorean Theorem

The Pythagorean Theorem is a special property of right triangles that has been used since ancient times. It is named after the Greek philosopher and mathematician Pythagoras who lived around $500$ BCE.

Remember that a right triangle has a $90$ ° angle, which we usually mark with a small square in the corner. The side of the triangle opposite the $90$ ° angle is called the hypotenuse, and the other two sides are called the legs. See (Figure 8).

In a right triangle, the side opposite the $90$ ° angle is called the hypotenuse and each of the other sides is called a leg.

Three right triangles are shown. Each has a box representing the right angle. The first one has the right angle in the lower left corner, the next in the upper left corner, and the last one at the top. The two sides touching the right angle are labeled “leg” in each triangle. The sides across from the right angles are labeled “hypotenuse.”

Figure 8

The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. It states that in any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse.

The Pythagorean Theorem

In any right triangle $mathematical expression$ ,

$a to the 2+b to the 2=c to the 2$

where $c$ is the length of the hypotenuse $a$ and $b$ are the lengths of the legs.

A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked a and b.

To solve problems that use the Pythagorean Theorem, we will need to find square roots. We defined the notation $the square root of m$ in this way:

$mathematical expression$

For example, we found that $the square root of 25$ is $5$ because $5 to the 2=25$ .

We will use this definition of square roots to solve for the length of a side in a right triangle.

EXAMPLE 7

Use the Pythagorean Theorem to find the length of the hypotenuse.

Right triangle with legs labeled as 3 and 4.

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the length of the hypotenuse of the triangle
Step 3. Name. Choose a variable to represent it.	Let $c=the length of the hypotenuse$
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The length of the hypotenuse is 5.

TRY IT 7.1

Use the Pythagorean Theorem to find the length of the hypotenuse.

A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked 6 and 8.

Show answer

TRY IT 7.2

Use the Pythagorean Theorem to find the length of the hypotenuse.

A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as c. One of the sides touching the right angle is labeled as 15, the other is labeled “8”.

Show answer

EXAMPLE 8

Use the Pythagorean Theorem to find the length of the longer leg.

Right triangle is shown with one leg labeled as 5 and hypotenuse labeled as 13.

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	The length of the leg of the triangle
Step 3. Name. Choose a variable to represent it.	Let $b=the leg of the triangle$ Label side b
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation. Isolate the variable term. Use the definition of the square root. Simplify.
Step 6. Check:
Step 7. Answer the question.	The length of the leg is 12.

TRY IT 8.1

Use the Pythagorean Theorem to find the length of the leg.

A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 15, the other is labeled “b”.

Show answer

TRY IT 8.2

Use the Pythagorean Theorem to find the length of the leg.

A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 15. One of the sides touching the right angle is labeled as 9, the other is labeled “b”.

Show answer

EXAMPLE 9

Kelvin is building a gazebo and wants to brace each corner by placing a $10-inch$ wooden bracket diagonally as shown. How far below the corner should he fasten the bracket if he wants the distances from the corner to each end of the bracket to be equal? Approximate to the nearest tenth of an inch.

A picture of a gazebo is shown. Beneath the roof is a rectangular shape. There are two braces from the top to each side. The brace on the left is labeled as 10 inches. From where the brace hits the side to the roof is labeled as x.

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the distance from the corner that the bracket should be attached
Step 3. Name. Choose a variable to represent it.	Let x = the distance from the corner
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation. Isolate the variable. Use the definition of the square root. Simplify. Approximate to the nearest tenth.
Step 6. Check: Yes.
Step 7. Answer the question.	Kelvin should fasten each piece of wood approximately 7.1″ from the corner.

TRY IT 9.1

John puts the base of a $13-ft$ ladder $5$ feet from the wall of his house. How far up the wall does the ladder reach?

A picture of a house is shown. There is a ladder leaning against the side of the house. The ladder is labeled 13 feet. The horizontal distance from the ladder's base to the house is labeled 5 feet.

Show answer

12 feet

TRY IT 9.2

Randy wants to attach a $17-ft$ string of lights to the top of the $15-ft$ mast of his sailboat. How far from the base of the mast should he attach the end of the light string?

A picture of a boat is shown. The height of the centre pole is labeled 15 feet. The string of lights is at a diagonal from the top of the pole and is labeled 17 feet.

Show answer

8 feet

Key Concepts

Supplementary and Complementary Angles
- If the sum of the measures of two angles is 180°, then the angles are supplementary.
- If $mathematical expression$ and $mathematical expression$ are supplementary, then $mathematical expression$ .
- If the sum of the measures of two angles is 90°, then the angles are complementary.
- If $mathematical expression$ and $mathematical expression$ are complementary, then $mathematical expression$ .
Solve Geometry Applications
1. Read the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information.
2. Identify what you are looking for.
3. Name what you are looking for and choose a variable to represent it.
4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
5. Solve the equation using good algebra techniques.
6. Check the answer in the problem and make sure it makes sense.
7. Answer the question with a complete sentence.
Sum of the Measures of the Angles of a Triangle
- For any $mathematical expression$ , the sum of the measures is 180°
- $mathematical expression$
Right Triangle
- A right triangle is a triangle that has one 90° angle, which is often marked with a ⦜ symbol.
Properties of Similar Triangles
- If two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths have the same ratio.

Glossary

angle: An angle is formed by two rays that share a common endpoint. Each ray is called a side of the angle.

complementary angles: If the sum of the measures of two angles is $90$ °, then they are called complementary angles.

hypotenuse: The side of the triangle opposite the 90° angle is called the hypotenuse.

legs of a right triangle: The sides of a right triangle adjacent to the right angle are called the legs.

right triangle: A right triangle is a triangle that has one $90$ ° angle.

similar figures: In geometry, if two figures have exactly the same shape but different sizes, we say they are similar figures.

supplementary angles: If the sum of the measures of two angles is $180$ °, then they are called supplementary angles.

triangle: A triangle is a geometric figure with three sides and three angles.

vertex of an angle: When two rays meet to form an angle, the common endpoint is called the vertex of the angle.

Practice Makes Perfect

Use the Properties of Angles

In the following exercises, find a) the supplement and b) the complement of the given angle.

1. $53$ °	2. $16$ °
3. $29$ °	4. $72$ °

In the following exercises, use the properties of angles to solve.

5. Find the supplement of a $135$ ° angle.	6. Find the complement of a $38$ ° angle.
7. Find the complement of a $27.5$ ° angle.	8. Find the supplement of a $109.5$ ° angle.
9. Two angles are supplementary. The larger angle is $56$ ° more than the smaller angle. Find the measures of both angles.	10. Two angles are supplementary. The smaller angle is $36$ ° less than the larger angle. Find the measures of both angles.
11. Two angles are complementary. The smaller angle is $34$ ° less than the larger angle. Find the measures of both angles.	12. Two angles are complementary. The larger angle is $52$ ° more than the smaller angle. Find the measures of both angles.

Use the Properties of Triangles

In the following exercises, solve using properties of triangles.

13. The measures of two angles of a triangle are $26$ ° and $98$ °. Find the measure of the third angle.	14. The measures of two angles of a triangle are $61$ ° and $84$ °. Find the measure of the third angle.
15. The measures of two angles of a triangle are $105$ ° and $31$ °. Find the measure of the third angle.	16. The measures of two angles of a triangle are $47$ ° and $72$ °. Find the measure of the third angle.
17. One angle of a right triangle measures $33$ °. What is the measure of the other angle?	18. One angle of a right triangle measures $51$ °. What is the measure of the other angle?
19. One angle of a right triangle measures $22.5$ °. What is the measure of the other angle?	20. One angle of a right triangle measures $36.5$ °. What is the measure of the other angle?
21. The two smaller angles of a right triangle have equal measures. Find the measures of all three angles.	22. The measure of the smallest angle of a right triangle is $20$ ° less than the measure of the other small angle. Find the measures of all three angles.
23. The angles in a triangle are such that the measure of one angle is twice the measure of the smallest angle, while the measure of the third angle is three times the measure of the smallest angle. Find the measures of all three angles.	24. The angles in a triangle are such that the measure of one angle is $20$ ° more than the measure of the smallest angle, while the measure of the third angle is three times the measure of the smallest angle. Find the measures of all three angles.

Find the Length of the Missing Side

In the following exercises, $mathematical expression$ is similar to $mathematical expression$ . Find the length of the indicated side.

25. side $b$

26. side $x$

On a map, San Francisco, Las Vegas, and Los Angeles form a triangle whose sides are shown in the figure below. The actual distance from Los Angeles to Las Vegas is $270$ miles.

A triangle is shown. The vertices are labeled San Francisco, Las Vegas, and Los Angeles. The side across from San Francisco is labeled 1 inch, the side across from Las Vegas is labeled 1.3 inches, and the side across from Los Angeles is labeled 2.1 inches.

27. Find the distance from Los Angeles to San Francisco.

28. Find the distance from San Francisco to Las Vegas.

Use the Pythagorean Theorem

In the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse.

29.	30.
31.	32.

Find the Length of the Missing Side

In the following exercises, use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary.

33.	34.
35.	36.
37.	38.
39.	40.

In the following exercises, solve. Approximate to the nearest tenth, if necessary.

41. A $13-foot$ string of lights will be attached to the top of a $12-foot$ pole for a holiday display. How far from the base of the pole should the end of the string of lights be anchored?

A vertical pole is shown with a string of lights going from the top of the pole to the ground. The pole is labeled 12 feet. The string of lights is labeled 13 feet.

42. Pam wants to put a banner across her garage door to congratulate her son on his college graduation. The garage door is $12$ feet high and $16$ feet wide. How long should the banner be to fit the garage door?

A picture of a house is shown. The rectangular garage is 12 feet high and 16 feet wide. A blue banner goes diagonally across the garage.

43. Chi is planning to put a path of paving stones through her flower garden. The flower garden is a square with sides of $10$ feet. What will the length of the path be?

A square garden is shown. One side is labeled as 10 feet. There is a diagonal path of blue circular stones going from the lower left corner to the upper right corner.

44. Brian borrowed a $20-foot$ extension ladder to paint his house. If he sets the base of the ladder $6$ feet from the house, how far up will the top of the ladder reach?

A picture of a house is shown with a ladder leaning against it. The ladder is labeled 20 feet tall. The horizontal distance from the house to the base of the ladder is 6 feet.

Everyday Math

45. Building a scale model Joe wants to build a doll house for his daughter. He wants the doll house to look just like his house. His house is $30$ feet wide and $35$ feet tall at the highest point of the roof. If the dollhouse will be $2.5$ feet wide, how tall will its highest point be?

46. Measurement A city engineer plans to build a footbridge across a lake from point $X$ to point $Y$ , as shown in the picture below. To find the length of the footbridge, she draws a right triangle $XYZ$ , with right angle at $X$ . She measures the distance from $X$ to $Z,800$ feet, and from $Y$ to $Z,1,000$ feet. How long will the bridge be?

A lake is shown. Point Y is on one side of the lake, directly across from point X. Point Z is on the same side of the lake as point X.

Writing Exercises

47. Write three of the properties of triangles from this section and then explain each in your own words.

48. Explain how the figure below illustrates the Pythagorean Theorem for a triangle with legs of length $3$ and $4$ .

Three squares are shown, forming a right triangle in the center. Each square is divided into smaller squares. The smallest square is divided into 9 small squares. The medium square is divided into 16 small squares. The large square is divided into 25 small squares.

Answers

1. a) 127° b) 37°	3. a) 151° b) 61°	5. 45°
7. 62.5°	9. 62°, 118°	11. 62°, 28°
13. 56°	15. 44°	17. 57°
19. 67.5°	21. 45°, 45°, 90°	23. 30°, 60°, 90°
25. 12	27. 351 miles	29. 15
31. 25	33. 8	35. 12
37. 10.2	39. 8	41. 5 feet
43. 14.1 feet	45. 2.9 feet	47. Answers will vary.

Attributions

This chapter has been adapted from “Use Properties of Angles, Triangles, and the Pythagorean Theorem” in Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

9.2 Solve Applications: Sine, Cosine and Tangent Ratios.

Learning Objectives

By the end of this section, you will be able to:

Find missing side of a right triangle using sine, cosine, or tangent ratios
Find missing angle of a right triangle using sine, cosine, or tangent ratios
Solve applications using right angle trigonometry

Now, that we know the fundamentals of algebra and geometry associated with a right triangle, we can start exploring trigonometry. Many real life problems can be represented and solved using right angle trigonometry.

Sine, Cosine, and Tangent Ratios

We know that any right triangle has three sides and a right angle. The side opposite to the right angle is called the hypotenuse. The other two angles in a right triangle are acute angles (with a measure less than 90 degrees). One of those angles we call reference angle and we use θ (theta) to represent it.

The hypotenuse is always the longest side of a right triangle. The other two sides are called opposite side and adjacent side. The names of those sides depends on which of the two acute angles is being used as a reference angle.

Figure 1.

In the right triangle each side is labeled with a lowercase letter to match the uppercase letter of the opposite vertex.

EXAMPLE 1

Label the sides of the triangle and find the hypotenuse, opposite, and adjacent.

Solution

We labeled the sides with a lowercase letter to match the uppercase letter of the opposite vertex.

c is hypotenuse

a is opposite

b is adjacent

TRY IT 1.1

Label the sides of the triangle and find the hypotenuse, opposite and adjacent.

Show answer

y is hypotenuse

z is opposite

x is adjacent

TRY IT 1.2

Label the sides of the triangle and find the hypotenuse, opposite and adjacent.

Show answer

r is hypotenuse

t is opposite

s is adjacent

Trigonometric Ratios

Trigonometric ratios are the ratios of the sides in the right triangle. For any right triangle we can define three basic trigonometric ratios: sine, cosine, and tangent.

Let us refer to Figure 1 and define the three basic trigonometric ratios as:

Three Basic Trigonometric Ratios

sine θ = $the length of the opposite side over the length of the hypotenuse side$
cosine θ = $the length of the adjacent side over the length of the hypotenuse side$
tangent θ = $the length of the opposite side over the length of the adjacent side$

Where θ is the measure of a reference angle measured in degrees.

Very often we use the abbreviations for sine, cosine, and tangent ratios.

sin θ = $opp over hyp$
cos θ = $adj over hyp$
tan θ = $opp over adj$

Some people remember the definition of the trigonometric ratios as SOH CAH TOA.

Let’s use the $mathematical expression$ from Example 1 to find the three ratios.

EXAMPLE 2

For the given triangle find the sine, cosine and tangent ratio.

Solution

sin θ = $f over d$

cos θ = $e over d$

tan θ = $f over e$

TRY IT 2.1

For the given triangle find the sine cosine and tangent ratio.

Show answer

sin θ = $z over y$

cos θ = $x over y$

tan θ = $z over x$

TRY IT 2.2

For the given triangle find the sine, cosine and tangent ratio.

Show answer

sin θ = $t over r$

cos θ = $s over r$

tan θ = $t over s$

In Example 2, our reference angles can be $mathematical expression$ or $mathematical expression$ . Using the definition of trigonometric ratios, we can write sinE= $e over d$ , cosE= $f over d$ , and tanE= $e over f$ .

When calculating we will usually round the ratios to four decimal places and at the end our final answer to one decimal place unless stated otherwise.

EXAMPLE 3

For the given triangle find the sine, cosine and tangent ratios. If necessary round to four decimal places.

Solution

We have two possible reference angles: R an S.

Using the definitions, the trigonometric ratios for angle R are:

sin R= $4 over 5$ = 0.8
cos R= $3 over 5$ = 0.6
tan R= $4 over 3$ = 1.3333…

Using the definitions, the trigonometric ratios for angle S:

sin S = $3 over 5$ = 0.6
cos S = $4 over 5$ = 0.8
tan S = $3 over 4$ = 0.75

TRY IT 3.1

For the given triangle find the sine, cosine, and tangent ratios. If necessary round to four decimal places.

Show answer

sin F = $8 over 10$ = 0.8
cos F = $6 over 10$ =0.6
tan F = $8 over 6$ = 1.3333…
sin D = $6 over 10$ = 0.6
cos D = $8 over 10$ = 0.8
tan D = $6 over 8$ = 0.75

TRY IT 3.2

For given triangle find the sine, cosine and tangent ratios. If necessary round to four decimal places.

Show answer

sin A = $5 over 5.8$ = 0.8621
cos A = $3 over 5.8$ =0.5172
tan A = $5 over 3$ = 1.6667
sin C = $3 over 5.8$ = 0.5172
cos C = $5 over 5.8$ = 0.8621
tan C = $3 over 5$ = 0.6

Now, let us use a scientific calculator to find the trigonometric ratios. Can you find the sin, cos, and tan buttons on your calculator? To find the trigonometric ratios make sure your calculator is in Degree Mode.

EXAMPLE 4

Using a calculator find the trigonometric ratios. If necessary, round to 4 decimal places.

a) sin 30°

b) cos 45°

c) tan 60°

Solution

Make sure your calculator is in Degree Mode.

a) Using a calculator find that sin 30° = 0.5

b) Using a calculator find that cos 45° = 0.7071 Rounded to 4 decimal places.

c) Using a calculator find that tan 60° = 1.7321 Rounded to 4 decimal places.

TRY IT 4.1

Find the trigonometric ratios. If necessary, round to 4 decimal places.

a) sin 60°

b) cos 30°

c) tan 45°

Show answer

a) sin 60° = 0.8660

b) cos 30° = 0.8660

c) tan 45° = 1

TRY IT 4.2

Find the trigonometric ratios. If necessary, round to 4 decimal places.

a) sin 35°

b) cos 67°

c) tan 83°

Show answer

a) sin 35° = 0.5736

b) cos 67 ° = 0.3907

c) tan 83° = 8.1443

Finding Missing Sides of a Right Triangle

In this section you will be using trigonometric ratios to solve right triangle problems. We will adapt our problem solving strategy for trigonometry applications. In addition, since those problems will involve the right triangle, it is helpful to draw it (if the drawing is not given) and label it with the given information.We will include this in the first step of the problem solving strategy for trigonometry applications.

HOW TO: Solve Trigonometry Applications

Read the problem and make sure all the words and ideas are understood. Draw the right triangle and label the given parts.
Identify what we are looking for.
Label what we are looking for by choosing a variable to represent it.
Find the required trigonometric ratio.
Solve the ratio using good algebra techniques.
Check the answer by substituting it back into the ratio in step 4 and by making sure it makes sense in the context of the problem.
Answer the question with a complete sentence.

In the next few examples, having given the measure of one acute angle and the length of one side of the right triangle, we will solve the right triangle for the missing sides.

EXAMPLE 5

Find the missing sides. Round your final answer to two decimal places

Solution

1. Read the problem and make sure all the words and ideas are understood. Draw the right triangle and label the given parts.	A drawing is given. Angle Y is our reference angle, y is opposite side, z is adjacent side, and x=14 is the hypotenuse.
2. Identify what we are looking for.	a) the opposite side	b) adjacent side
3.Label what we are looking for by choosing a variable to represent it.	y=?	z=?
4. Find the required trigonometric ratio.	sin 35° = $y over 14$	cos 35° = $z over 14$
5. Solve the ratio using good algebra techniques.	14 sin 35° = y 8.03 = y	14 cos 35° = z 11.47 = z
6. Check the answer in the problem and by making sure it makes sense.	0.57 $=$ 8.03 $divided by$ 14 0.57 = 0.57 $mathematical expression$	0.82 $=$ 11.47 $divided by$ 14 0.82 = 0.82 $mathematical expression$
7. Answer the question with a complete sentence.	The opposite side is 8.03	The adjacent side is 11.47

TRY IT 5.1

Find the missing sides. Round your final answer to one decimal place.

Show answer

a = 20.2

b = 16.4

TRY IT 5.2

Find the missing sides. Round your final answer to one decimal place.

Show answer

d = 3.4

f = 9.4

EXAMPLE 6

Find the hypotenuse. Round your final answer to one decimal place.

Solution

1. Read the problem and make sure all the words and ideas are understood. Draw the right triangle and label the given parts.

A drawing is given. Angle S is our reference angle, s is opposite side, r = 4 is the adjacent side, and p is the hypotenuse

2. Identify what we are looking for.

the hypotenuse

3.Label what we are looking for by choosing a variable to represent it.

p=?

4. Find the required trigonometric ratio.

cos 32° = $4 over p$

5. Solve the ratio using good algebra techniques.

0.8480 = $4 over p$

p = 4.7170

Rounding the ratios to 4 decimal places

6. Check the answer in the problem and by making sure it makes sense.

0.8480 $=$ $4 over 4.7170$

0.8480 = 0.8480 $mathematical expression$

7. Answer the question with a complete sentence.

The hypotenuse is 4.7

Round my final answer to one decimal place.

TRY IT 6.1..

Find the hypotenuse. Round your final answer to one decimal place.

Show answer

p = 22.7

TRY IT 6.2

Find the hypotenuse. Round your final answer to one decimal place.

Show answer

p = 6.5

Finding Missing Angles of a Right Triangle

Sometimes we have a right triangle with only the sides given. How can we find the missing angles? To find the missing angles, we use the inverse of the trigonometric ratios. The inverse buttons sin^-1, cos^-1, and tan^-1 are on your scientific calculator.

EXAMPLE 7

Find the angles. Round your final answer to one decimal place.

a) sin A = 0.5

b) cos B = 0.9735

c) tan C = 2.89358

Solution

Use your calculator and press the 2nd FUNCTION key and then press the SIN, COS, or TAN key

a) A = sin^-10.5

$mathematical expression$ = 30°

b) B = cos^-10.9735

$mathematical expression$ = 13.2° Rounded to one decimal place

c) C = tan^-12.89358

$mathematical expression$ = 70.9° Rounded to one decimal place

TRY IT 7.1

Find the angles. Round your final answer to one decimal place.

a) sin X = 1

b) cos Y = 0.375

c) tan Z = 1.676767

Show answer

a) $mathematical expression$ = 90°

b) $mathematical expression$ = 68°

c) $mathematical expression$ = 59.2°

TRY IT 7.2

Find the angles. Round your final answer to one decimal place.

a) sin C = 0

b) cos D = 0.95

c) tan F = 6.3333

Show answer

a) $mathematical expression$ = 0°

b) $mathematical expression$ = 18.2°

c) $mathematical expression$ = 81°

In the example below we have a right triangle with two sides given. Our acute angles are missing. Let us see what the steps are to find the missing angles.

EXAMPLE 8

Find the missing $mathematical expression$ . Round your final answer to one decimal place.

Solution

1. Read the problem and make sure all the words and ideas are understood. Draw the right triangle and label the given parts.

A drawing is given. Angle T is our reference angle, t = 7 is the opposite side, s is adjacent side, and r =11 is the hypotenuse

2. Identify what we are looking for.

angle T

3.Label what we are looking for by choosing a variable to represent it.

$mathematical expression$ =?

4. Find the required trigonometric ratio.

sin T = $7 over 11$

5. Solve the ratio using good algebra techniques.

sin T = 0.6364

T = sin^-10.6364

$mathematical expression$ = 39.5239°

6. Check the answer in the problem and by making sure it makes sense.

sin 39.5239° $=$ 0.6364

0.6364 = 0.6364 $mathematical expression$

7. Answer the question with a complete sentence.

The missing angle T is 39.5°.

TRY IT 8.1

Find the missing angle X. Round your final answer to one decimal place.

Show answer

20.1°

TRY IT 8.2

Find the missing angle Z. Round your final answer to one decimal place.

Show answer

69.9°

EXAMPLE 9

Find the missing angle A. Round your final answer to one decimal place.

Solution

1. Read the problem and make sure all the words and ideas are understood. Draw the right triangle and label the given parts.

A drawing is given. Angle A is our reference angle, a = 9 is the opposite side, c = 5 is the adjacent side, and b is the hypotenuse

2. Identify what we are looking for.

angle A

3.Label what we are looking for by choosing a variable to represent it.

$mathematical expression$ =?

4. Find the required trigonometric ratio.

tan A = $9 over 5$

5. Solve the ratio using good algebra techniques.

tan A = 1.8

A = tan^-1 1.8

$mathematical expression$ = 60.9°

6. Check the answer in the problem and by making sure it makes sense.

tan 60.9° $=$ 1.8

1.8 = 1.8 $mathematical expression$

7. Answer the question with a complete sentence.

The missing angle A is 60.9°.

TRY IT 9.1

Find the missing angle C. Round your final answer to one decimal place.

Show answer

29.1°

TRY IT 9.2

Find the missing angle E. Round your final answer to one decimal place.

Show answer

36.9°

Solving a Right Triangle

From the section before we know that any triangle has three sides and three interior angles. In a right triangle, when all six parts of the triangle are known, we say that the right triangle is solved.

EXAMPLE 10

Solve the right triangle. Round your final answer to one decimal place.

Solution

Since the sum of angles in any triangle is 180°, the measure of angle B can be easy calculated.

$mathematical expression$ = 180° − 90° − 42°

$mathematical expression$ = 48°

1. Read the problem and make sure all the words and ideas are understood. Draw the right triangle and label the given parts.	A drawing is given. Angle A is our reference angle, a = 8 is the opposite side, b is the adjacent side, and c is the hypotenuse.
2. Identify what we are looking for.	a) adjacent side	b) hypotenuse
3.Label what we are looking for by choosing a variable to represent it.	b = ?	c = ?
4. Find the required trigonometric ratio.	tan 42° = $8 over b$	sin 42° = $8 over c$
5. Solve the ratio using good algebra techniques.	0.9004 = $8 over b$ 0.9004 b = 8 b = 8.8849	0.6691 = $8 over c$ 0.6691 c = 8 c = 11.9563
6. Check the answer in the problem and by making sure it makes sense.	tan 42 ° $=$ $8 over 8.8849$ 0.9 = 0.9 $mathematical expression$	sin 42° $=$ $8 over 11.9563$ 0.6691 = 0.6691 $mathematical expression$
7. Answer the question with a complete sentence.	The adjacent side is 8.9. Rounded to one decimal place.	The hypotenuse is 12

We solved the right triangle

$mathematical expression$ = 42°

$mathematical expression$ = 48°

$mathematical expression$ = 90°

a = 8

b = 8.9

c = 12

TRY IT 10.1

Solve the right triangle. Round your final answer to one decimal place.

$mathematical expression$ = 21°

$mathematical expression$ = 69°

$mathematical expression$ = 90°

Show answer

a = 6

b = 15.6

c = 16.7

TRY IT 10.2

Solve the right triangle. Round your final answer to one decimal place.

$mathematical expression$ =16°

$mathematical expression$ = 74°

$mathematical expression$ = 90°

Show answer

a = 2.9

b = 10

c = 10.4

EXAMPLE 11

Solve the right triangle. Round to two decimal places.

Solution

1. Read the problem and make sure all the words and ideas are understood. Draw the right triangle and label the given parts.	A drawing is given. Let angle D be our reference angle, d = 4 is the opposite side, f is the adjacent side, and e = 9 is the hypotenuse
2. Identify what we are looking for.	a) angle D	b) adjacent
3.Label what we are looking for by choosing a variable to represent it.	$mathematical expression$ =?	f = ?
4. Find the required trigonometric ratio.	sin D = $4 over 9$	4² + f² = 9²
5. Solve the ratio using good algebra techniques.	sin D = 0.4444 D = sin^-10.4444 $mathematical expression$ = 26.3850°	16 + f² = 81 f² = 81 – 16 f² = 65 f = square root of 65 f = 8.06
6. Check the answer in the problem and by making sure it makes sense.	sin 26.3850° $=$ $4 over 9$ 0.4444 =0.4444 $mathematical expression$	4² + 8.06² $=$ 9² 81 = 81 $mathematical expression$
7. Answer the question with a complete sentence.	The missing angle D is 26.39°.	The adjacent side is 8.06 Rounded to two decimal places

The missing angle F = 180° – 90° – 26.39° = 63.64°

We solved the right triangle

$mathematical expression$ = 26.39°

$mathematical expression$ = 90°

$mathematical expression$ = 63.61°

d = 4

e = 9

f = 8.06

TRY IT 11.1

Solve the right triangle. Round to one decimal place.

$mathematical expression$ = 29.3°

$mathematical expression$ = 90°

$mathematical expression$ = 60.7°

Show answer

d = 29.4

e = 18.4

f = 60.6

TRY IT 11.2

Solve the right triangle. Round to one decimal place.

$mathematical expression$ = 45.6°

$mathematical expression$ = 90°

$mathematical expression$ = 44.4°

Show answer

d = 7.1

e = 10

f = 7

Solve Applications Using Trigonometric Ratios

In the previous examples we were able to find missing sides and missing angles of a right triangle. Now, let’s use the trigonometric ratios to solve real-life problems.

Many applications of trigonometric ratios involve understanding of an angle of elevation or angle of depression.

The angle of elevation is an angle between the horizontal line (ground) and the observer’s line of sight.

The angle of depression is the angle between horizontal line (that is parallel to the ground) and the observer’s line of sight.

EXAMPLE 12

James is standing 31 metres away from the base of the Harbour Centre in Vancouver. He looks up to the top of the building at a 78° angle. How tall is the Harbour Centre?

Solution

1. Read the problem and make sure all the words and ideas are understood. Draw the right triangle and label the given parts.

Angle X is our reference angle, x is opposite side, y = 31 m is the adjacent side, and z is the hypotenuse.

2. Identify what we are looking for.

The opposite side

3.Label what we are looking for by choosing a variable to represent it.

x=?

4. Find the required trigonometric ratio.

tan 78° = $x over 31$

5. Solve the ratio using good algebra techniques.

4.7046 = $x over 31$

x = 145.8426

6. Check the answer in the problem and by making sure it makes sense.

4.7046 $=$ $145.8426 over 31$

4.7046 = 4.7046 $mathematical expression$

7. Answer the question with a complete sentence.

The Harbour Centre is 145.8426 metres or rounded to 146 metres.

TRY IT 12.1

Nicole is standing 75 feet away from the base of the Living Shangri-La, the tallest building in British Columbia. She looks up to the top of the building at a 83.5° angle. How tall is the Living Shangri-La?

Show answer

658.3 feet.

TRY IT 12.2

Kelly is standing 23 metres away from the base of the tallest apartment building in Prince George and looks at the top of the building at a 62° angle. How tall is the building?

Show answer

43.3 metres

EXAMPLE 13

Thomas is standing at the top of the building that is 45 metres high and looks at his friend that is standing on the ground, 22 metres from the base of the building. What is the angle of depression?

Solution

1. Read the problem and make sure all the words and ideas are understood. Draw the right triangle and label the given parts.

Angle Y is our reference angle, y = 45 m is the opposite side, z = 22 m is the adjacent side, and x is the hypotenuse

2. Identify what we are looking for.

angle Y

3.Label what we are looking for by choosing a variable to represent it.

$mathematical expression$ =?

4. Find the required trigonometric ratio.

tan Y = $45 over 22$

5. Solve the ratio using good algebra techniques.

tan Y = 2.0455

Y = tan ^–¹2.0455

$mathematical expression$ = 63.9470°

6. Check the answer in the problem and by making sure it makes sense.

tan 63.9470° $=$ 2.0455

2.0455 = 2.0455 $mathematical expression$

7. Answer the question with a complete sentence.

The angle of depression is 63.9470° or 64° rounded to one decimal place.

TRY IT 13.1

Hemanth is standing on the top of a cliff 250 feet above the ground and looks at his friend that is standing on the ground, 40 feet from the base of the cliff. What is the angle of depression?

Show answer

80.9°

TRY IT 13.2

Klaudia is standing on the ground, 25 metres from the base of the cliff and looks up at her friend on the top of a cliff 100 metres above the ground. What is the angle of elevation?

Show answer

76°

Key Concepts

Three Basic Trigonometric Ratios: (Where θ is the measure of a reference angle measured in degrees.)
- sine θ = $the length of the opposite side over the length of the hypotenuse side$
- cosine θ = $the length of the adjacent side over the length of the hypotenuse side$
- tangent θ = $the length of the opposite side over the length of the adjacent side$
Problem-Solving Strategy for Trigonometry Applications
1. Read the problem and make sure all the words and ideas are understood. Draw the right triangle and label the given parts.
2. Identify what we are looking for.
3. Label what we are looking for by choosing a variable to represent it.
4. Find the required trigonometric ratio.
5. Solve the ratio using good algebra techniques.
6. Check the answer by substituting it back into the ratio solved in step 5 and by making sure it makes sense in the context of the problem.
7. Answer the question with a complete sentence.

Practice Makes Perfect

Label the sides of the triangle.

1	2.
3. If the reference angle in Question 1 is B, Find the adjacent ?	4. If the reference angle in Question 2 is Z, find the opposite ?

Label the sides of the triangle and find the hypotenuse, opposite and adjacent.

Use your calculator to find the given ratios. Round to four decimal places if necessary:

7. $mathematical expression$	8. $mathematical expression$
9. $mathematical expression$	10. $mathematical expression$

For the given triangles, find the sine, cosine and tangent of the θ.

11.	12.
13.	14.

For the given triangles, find the missing side. Round it to one decimal place.

15. Find the hypotenuse.	16. Find b if a = 6.
17. Find the opposite.	18. Find the adjacent.

For the given triangles, find the missing sides. Round it to one decimal place.

19.

20.

Solve the triangles. Round to one decimal place.

21.	22.
23.	24.
25. A surveyor stands 75 metres from the bottom of a tree and looks up at the top of the tree at a 48° angle. How tall is the tree?	26. A tree makes a shadow that is 6 metres long when the angle of elevation to the sun is 52°. How tall is the tree?
27. A ladder that is 15 feet is leaning against a house and makes a 45° angle with the ground. How far is the base of the ladder from the house?	28. Matt is flying a kite and has let out 100 feet of string. The angle of elevation with the ground is 38°. How high is his kite above the ground?
29. Marta is flying a kite and has let out 28 metres of string. If the kite is 10 metres above the ground, what is the angle of elevation?	30. An airplane takes off from the ground at the angle of 25°. If the airplane traveled 200 kilometres, how high above the ground is it?

Answers

1.	3. c	5. g is opposite , f is adjacent, and e is hypotenuse
7. 0.7314	9. 0.2126	11. sin θ = $g over e$ , cos θ = $f over e$ , tan θ = $g over f$
13. sin θ = $s over r$ , cos θ = $t over r$ , tan θ = $s over t$	15. b = 19.8	17. c = 12
19. y = 19.3, z = 8.2	21. $mathematical expression$ = 61° $mathematical expression$ = 29° $mathematical expression$ = 90° b = 38.5 c = 21.3 d = 44	23. $mathematical expression$ = 36.9° $mathematical expression$ = 90° $mathematical expression$ = 53.1° t = 15 r = 25 s = 20
25. 83.3 m	27. 10.6 ft	29. 20.9°

9.3 Chapter Review

Review Exercises

Use Properties of Angles

In the following exercises, solve using properties of angles.

1. What is the supplement of a $48$ ° angle?	2. What is the complement of a $61$ ° angle?
3. Two angles are complementary. The smaller angle is $24$ ° less than the larger angle. Find the measures of both angles.	4. Two angles are supplementary. The larger angle is $45$ ° more than the smaller angle. Find the measures of both angles.

Use Properties of Triangles

In the following exercises, solve using properties of triangles.

5. The measures of two angles of a triangle are $22$ and $85$ degrees. Find the measure of the third angle.	6. One angle of a right triangle measures $41.5$ degrees. What is the measure of the other small angle?
7. One angle of a triangle is $30$ ° more than the smallest angle. The largest angle is the sum of the other angles. Find the measures of all three angles.	8. One angle of a triangle is twice the measure of the smallest angle. The third angle is $60$ ° more than the measure of the smallest angle. Find the measures of all three angles.

In the following exercises, $mathematical expression$ is similar to $mathematical expression$ . Find the length of the indicated side.

Two triangles are shown. Triangle ABC is on the left. The side across from A is labeled 21, across from B is b, and across from C is 11.2. Triangle XYZ is on the right. The side across from X is labeled x, across from Y is 10, and across from Z is 8.

9. side $x$

10. side $b$

Use the Pythagorean Theorem

In the following exercises, use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary.

11.	12.
13.	14.
15.	16.

In the following exercises, solve. Approximate to the nearest tenth, if necessary.

17. Sergio needs to attach a wire to hold the antenna to the roof of his house, as shown in the figure. The antenna is $8$ feet tall and Sergio has $10$ feet of wire. How far from the base of the antenna can he attach the wire?

An image of a house is shown. A 10-foot wire is going from the roof of the house to the ground. The wire hits the house at a height of 8 feet.

18. Seong is building shelving in his garage. The shelves are $36$ inches wide and $15$ inches tall. He wants to put a diagonal brace across the back to stabilize the shelves, as shown. How long should the brace be?

A rectangular shelf is shown, with a diagonal drawn in from the lower left corner to the upper right corner. The side is labeled 15 inches, the top is labeled 36 inches.

Find missing side of a right triangle using sine, cosine, or tangent ratios.

19. Label the triangle and find the sine cosine and tangent of θ.

20. If reference angle in above triangle is angle T, label the triangle and find the sine, cosine, and tangent of T.

Find missing angle of a right triangle using sine, cosine, or tangent ratios.

21. Find angle M

22. Find angle L.

Solve the right triangle.

23. Solve the triangle.

24.

Solve applications using right angle trigonometry.

25. A 13-foot string of lights will be attached to the top of a 12-foot pole for a holiday display, as shown below. What is the angle that the string of lights makes with the ground?

A right triangle with one leg marked 12 and hypotenuse marked 13.

26. Brian borrowed a 20 foot extension ladder to use when he paints his house. If he sets the base of the ladder 6 feet from the house, as shown below, what is the angle that the ladder makes with the ground?

A house is shown with a ladder leaning against it. The ladder is marked 20’, and the distance from the house to the base of the ladder is marked 6’.

27. John puts the base of a 13-foot ladder five feet from the wall of his house as shown below. What is the angle between the top of the ladder and the house ?

A house is shown with a ladder leaning against it. The ladder is marked 13’, and the distance from the house to the base of the ladder is marked 5’.

28. The sun is at an angle of elevation of 35°. If Bob casts a shadow that is 6 ft long, how tall is Bob?

29. A 27 foot guy wire to a pole makes an angle of 63.7° with the ground. How high from the ground is the wire attached to the pole?

30. A lighthouse is 20 metres tall. If the observer is looking at a boat that is 30 metres away from the base of the lighthouse, what is the angle of depression?

Review Answers

1. 132°	3. 33°, 57°	5. 73°
7. 30°, 60°, 90°	9. 15	11. 26
13. 8	15. 8.1	17. 6 feet
19. sin θ = $s over r$ , cos θ = $t over r$ , tan θ = $s over t$	21. 55.2°	23. $mathematical expression$ = 90° , $mathematical expression$ = 57° , $mathematical expression$ = 33°x = 14, y = 11.7, z= 7.6
25. 67.4°	27. 22.6°	29. 24

Practice Test

1. What is the supplement of a $mathematical expression$ ° angle?	2. Two angles are complementary. The smaller angle is $16$ ° less than the larger angle. Find the measures of both angles.
3. The measures of two angles of a triangle are $29$ and $75$ degrees. Find the measure of the third angle.	4. $mathematical expression$ is similar to $mathematical expression$ . Find the missing sides.
5. Use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary.	6. Find the hypotenuse.
7. Find angle G.	8. Solve the triangle.
9. The sun is at an angle 28°. If Adam casts a shadow that is 7 ft long, how tall is Adam?	10. The road rises 6 metres per every 100 horizontal metres. What is the angle of elevation.

Answers

1. 123°	2. 53°, 37°	3. 76°
4. b = 14, t = 7.5	5. b = 15.3	6. d = 18.4
7. $mathematical expression$ = 27°	8. $mathematical expression$ = 36.7°, $mathematical expression$ = 53.3°, $mathematical expression$ = 90°, c = 49, b = 65.7, d= 82	9. 5.5 ft
10. 3.4°

Acknowledgements

It is my genuine pleasure to express many thanks and gratitude to the people who have significantly contributed to my accomplishment:

Krista Lambert for her remarkable and endless support, advice, and encouragement throughout this project.
Josie Gray, Harper Friedman, and Kaitlyn Zheng for providing astounding technical support in completing, reviewing, and publishing this open textbook.
Hemanth Anil and Kimberly Lebel for their wonderful contribution, dedication, and time spent on formatting of the textbook.
Thompson Rivers University administration and employees for their ongoing support of Open Educational Resources.

Versioning History

This page provides a record of edits and changes made to this book since its initial publication. Whenever edits or updates are made in the text, we provide a record and description of those changes here. If the change is minor, the version number increases by 0.01. If the edits involve substantial updates, the version number increases to the next full number.

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Version	Date	Change	Details
1.01	March 26, 2021	Book published
1.02	May 19, 2021	Acknowledgements section added to the front matter.
1.03	November 17, 2021	Corrections made to Chapter 9.2	Solutions for TRY IT 11.1 altered, questions for Exercise 16 and 22 altered.
1.04	September 12, 2025	Corrections made to chapters 2.1 and 2.2	Solution for TRY IT 10.2 in chapter 2.1 was corrected. Questions for TRY IT 4.1 and TRY IT 8.1 in chapter 2.2 were altered.