Right Triangle Trigonometry - COPYRIGHTED MATERIAL

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[ 1CHAPTER

Right Triangle Trigonometry

To the ancient Greeks, trigonometry was the study of right triangles. Trigonometric functions (sine, cosine, tangent, cotangent, secant, and cosecant) can be defined as right triangle ratios (ratios of the lengths of sides of a right triangle). Thousands of years later, we still find applications of right triangle trigonometry today in sports, surveying, navigation,* and engineering.

*Section 1.5, Example 7 and Exercises 66?68 and 73?74.

LEARNING OBJECTIVES

Understand degree measure.

Learn the conditions that make two triangles similar.

Define the six trigonometric functions as ratios of lengths of the sides of right triangles.

Evaluate trigonometric functions exactly and with calculators.

Solve right triangles.

[IN THIS CHAPTER]

We will review angles, degree measure, and special right triangles. We will discuss the properties of similar triangles. We will use the concept of similar right triangles to define the six trigonometric functions as ratios of the lengths of the sides of right triangles (right triangle trigonometry).

RIGHT TRIANGLE TRIGONOMETRY

1.1

ANGLES, DEGREES, AND TRIANGLES

1.2

SIMILAR TRIANGLES

1.3

DEFINITION 1 OF TRIGONOMETRIC FUNCTIONS: RIGHT TRIANGLE RATIOS

1.4

EVALUATING TRIGONOMETRIC FUNCTIONS: EXACTLY AND WITH CALCULATORS

1.5

SOLVING RIGHT TRIANGLES

? Angles and Degree Measure

? Triangles ? Special Right

Triangles

? Finding Angle Measures Using Geometry

? Classification of Triangles

? Trigonometric Functions: Right Triangle Ratios

? Cofunctions

? Evaluating Trigonometric Functions Exactly for Special Angle Measures: 308, 458, and 608

? Using Calculators to Evaluate (Approximate) Trigonometric Function Values

? Representing Partial Degrees: DD or DMS

? Accuracy and Significant Digits

? Solving a Right Triangle Given an Acute Angle Measure and a Side Length

? Solving a Right Triangle Given the Lengths of Two Sides

3

4 CHAPTER 1 Right Triangle Trigonometry

1.1 ANGLES, DEGREES, AND TRIANGLES

SKILLS OBJECTIVES Find the complement and supplement of an angle. Use the Pythagorean theorem to solve a right triangle. Solve 308-608-908 and 458-458-908 triangles.

CONCEPTUAL OBJECTIVES

Understand that degrees are a measure of an angle. Understand that the Pythagorean theorem applies only to

right triangles. Understand that to solve a triangle means to find all of the

angle measures and side lengths.

1.1.1 S K I L L

Find the complement and supplement of an angle.

1.1.1 C O N C E P T U A L

Understand that degrees are a measure of an angle.

A

B

Line AB

A

B

Segment AB

A

B

Ray AB

Vertex

Angle

1.1.1 Angles and Degree Measure

The study of trigonometry relies heavily on the concept of angles. Before we define angles, let us review some basic terminology. A line is the straight path connecting two points (A and B) and extending beyond the points in both directions. The portion of the line between the two points (including the points) is called a line segment. A ray is the portion of the line that starts at one point (A) and extends to infinity (beyond B). Point A is called the endpoint of the ray.

In geometry, an angle is formed when two rays share the same endpoint. The common endpoint is called the vertex. In trigonometry, we say that an angle is formed when a ray is rotated around its endpoint. The ray in its original position is called the initial ray or the initial side of an angle. In the Cartesian plane (the rectangular coordinate plane), we usually assume the initial side of an angle is the positive x-axis and the vertex is located at the origin. The ray after it is rotated is called the terminal ray or the terminal side of an angle. Rotation in a counterclockwise direction corresponds to a positive angle, whereas rotation in a clockwise direction corresponds to a negative angle.

Terminal side

Positive angle Initial side

Initial side Negative angle

Terminal side

STUDY TIP

Positive angle: Counterclockwise Negative angle: Clockwise

[ ] CONCEPT CHECK

TRUE OR FALSE The measure of an acute angle is greater than the measure of an obtuse angle.

ANSWER False

Lengths, or distances, can be measured in different units: feet, miles, and meters are three common units. To compare angles of different sizes, we need a standard unit of measure. One way to measure the size of an angle is with degree measure. Wediscuss degrees now, and in Chapter 3, we discuss another angle measure called radians.

DEFINITION Degree Measure of Angles An angle formed by one complete counterclockwise rotation has measure 360 degrees, denoted 360?.

One complete revolution = 360?

Therefore,

a

counterclockwise

1 360

of

a

rotation

has

measure

1

degree.

WORDS

360? represents 1 complete rotation.

180?

represents

1 2

rotation.

90?

represents

1 4

rotation.

MATH

1 complete rotation 5 1 360? 5 360?

1 2

complete

rotation

5

1 2

360?

5

180?

1 4

complete

rotation

5

1 4

360?

5

90?

1.1 Angles, Degrees, and Triangles 5

The Greek letter u (theta) is the most common name for an angle in mathematics. Other common names for angles are a (alpha), b (beta), and g (gamma).

WORDS

An angle measuring exactly 90? is called a right angle. A right angle is often represented by the adjacent sides of a rectangle, indicating that the two rays are perpendicular.

MATH

= 90?

Right

angle:

1 4

rotation

STUDY TIP

Greek letters are often used to denote angles in trigonometry.

An angle measuring exactly 180? is called a straight angle.

= 180?

Straight

angle:

1 2

rotation

An angle measuring greater than 08, but less than 90?, is called an acute angle.

Acute angle 0? < < 90?

An angle measuring greater than 90?, but less than 180?, is called an obtuse angle.

Obtuse angle

90? < < 180?

If the sum of the measures of two positive angles is 90?, the angles are called complementary. We say that a is the complement of b (and vice versa).

If the sum of the measures of two positive angles is 180?, the angles are called supplementary. We say that a is the supplement of b (and vice versa).

Complementary angles + = 90?

Supplementary angles + = 180?

EXAMPLE 1 F inding Measures of Complementary and Supplementary Angles

Find the measure of each angle: a. Find the complement of 50?. b. Find the supplement of 110?. c. Represent the complement of a in terms of a. d.Find two supplementary angles such that the first angle is twice as large as the

second angle.

Solution: a. The sum of complementary angles is 90?.

Solve for u.

u 1 50? 5 90? u 5 40?

b. The sum of supplementary angles is 180?. Solve for u.

u 1 110? 5 180? u 5 70?

6 CHAPTER 1 Right Triangle Trigonometry

c. Let b be the complement of a. The sum of complementary angles is 90?. Solve for b.

d. The sum of supplementary angles is 180?. Let b 5 2a. Solve for a.

Substitute a 5 60? into b 5 2a.

a 1 b 5 90? b 5 90? 2 a

a 1 b 5 180? a 1 2a 5 180?

3a 5 180? a 5 60? b 5 120?

ANSWER

The angles have measures 45? and 135?.

1.1.2 S K I L L Use the Pythagorean theorem to solve a right triangle.

1.1.2 C O N C E P T U A L Understand that the Pythagorean theorem applies only to right triangles.

The angles have measures 60? and 120? .

Y O U R T U R N Find two supplementary angles such that the first angle is three times as large as the second angle.

1.1.2Triangles

Trigonometry originated as the study of triangles, with emphasis on calculations involving the lengths of sides and the measures of angles. Triangles are three-sided closed-plane figures. An important property of triangles is that the sum of the measures of the three angles of any triangle is 180?.

ANGLE SUM OF A TRIANGLE

The sum of the measures of the angles of any triangle is 180?.

+ + = 180?

ANSWER

68?

EXAMPLE 2 Finding an Angle of a Triangle If two angles of a triangle have measures 32? and 68?, what is the measure of the third angle?

32?

68?

Solution: The sum of the measures of all three angles is 180?. Solve for a.

32? 1 68? 1 a 5 180? a 5 80?

Y O U R T U R N If two angles of a triangle have measures 16? and 96?, what is the measure of the third angle?

1.1 Angles, Degrees, and Triangles 7

In geometry, some triangles are classified as equilateral, isosceles, and right. An equilateral triangle has three equal sides and therefore has three equal angles 16082. An isosceles triangle has two equal sides (legs) and therefore has two equal angles opposite those legs. Themost important triangle that we will discuss in this course is a right triangle. A right triangle is a triangle in which one of the angles is a right angle 190?2. Since one angle is 90?, the other two angles must be complementary 1sum to 90?2 so that the sum of all three angles is 180?. The longest side of a right triangle, called the hypotenuse, is opposite the right angle. The other two sides are called the legs of the right triangle.

Right triangle: Leg

Hypotenuse

Leg

The Pythagorean theorem relates the sides of a right triangle. It is important to note that length (a synonym of distance) is always positive.

STUDY TIP

In this book when we say "equal angles," this implies "equal angle measures." Similarly, when we say an angle is x8, this implies that the angle's measure is x8.

PYTHAGOREAN THEOREM

In any right triangle, the square of the length of the longest side (hypotenuse) is equal to the sum of the squares of the lengths of the other two sides (legs).

a2 1 b2 5 c2

a

c

[ ] CONCEPT CHECK

TRUE OR FALSE The hypotenuse is always longer than each of the legs of a right triangle.

ANSWER True

b

It is important to note that the Pythagorean theorem applies only to right triangles. In addition, it does not matter which leg is called a or b as long as the square of the longest side is equal to the sum of the squares of the smaller sides.

EXAMPLE 3 Using the Pythagorean Theorem to Find the Side of a Right Triangle

Suppose you have a 10-foot ladder and want to reach a height of 8 feet to clean out the gutters on your house. How far from the base of the house should the base of the ladder be?

Solution:

Label the unknown side as x.

10

8

10 ft

8 ft ?

Apply the Pythagorean theorem. Simplify.

x

x2 1 82 5 102 x2 1 64 5 100

8 CHAPTER 1 Right Triangle Trigonometry

ANSWER

15 ft

Solve for x. Length must be positive.

x2 5 36 x 5 66 x56

The ladder should be 6 feet from the base of the house along the ground.

Y O U R T U R N A steep ramp is being built for skateboarders. The height, horizontal ground distance, and ramp length form a right triangle. If the height is 9 feet and the horizontal ground distance is 12 feet, what is the length of the ramp?

When solving a right triangle exactly, simplification of radicals is often necessary. For example, if a side length of a triangle resulted in !17, the radical cannot be simplified any further. However, if a side length of a triangle resulted in !20, the radical would be simplified:

"20 5 "4 5 5 "4 "5 5 "22 "5 5 2"5

ANSWER

4"3

EXAMPLE 4 Using the Pythagorean Theorem with Radicals

Use the Pythagorean theorem to solve for the unknown side length in the given right triangle. Express your answer exactly in terms of simplified radicals.

Solution:

Apply the Pythagorean theorem.

32 1 x2 5 72

Simplify known squares. Solve for x.

9 1 x2 5 49 x2 5 40

7

x

x 5 6"40

3

The side length x is a distance that is positive. x 5 "40

Simplify the radical.

x 5 "4 10 5 "4 "10 5 2"10

f

2

Y O U R T U R N Use the Pythagorean theorem to solve for the

unknown side length in the given right triangle.

Express your answer exactly in terms of simplified radicals.

8 x

4

1.1.3 S K I L L Solve 308-608-908 and 458-458-908 triangles.

1.1.3 C O N C E P T U A L Understand that to solve a triangle means to find all of the angle measures and side lengths.

1.1.3 Special Right Triangles

Right triangles whose sides are in the ratios of 3-4-5, 5-12-13, and 8-15-17 are examples of right triangles that are special because their side lengths are equal to whole numbers that satisfy the Pythagorean theorem. A Pythagorean triple consists of three positive integers that satisfy the Pythagorean theorem.

32 1 42 5 52 52 1 122 5 132 82 1 152 5 172

There are two other special right triangles that warrant attention: a 308-608-908 triangle and a 458-45?-908 triangle. Although in trigonometry we focus more on the angles than

1.1 Angles, Degrees, and Triangles 9

on the side lengths, we are interested in special relationships between the lengths of the sides of these right triangles. We will start with a 458-458-908 triangle.

WORDS A 458-458-908 triangle is an isosceles (two legs are equal) right triangle.

Apply the Pythagorean theorem. Simplify the left side of the equation. Solve for the hypotenuse. The x and the hypotenuse are both lengths and, therefore, must be positive. This shows that the hypotenuse of a 458-458-908 is !2 times the length of either leg.

MATH

45? x

45? x

x2 1 x2 5 hypotenuse2 2x2 5 hypotenuse2

hypotenuse 5 6"2x2 5 6 !2 0 x 0

hypotenuse 5 !2x

45? 2x

x

45? x

If we let x 5 1, then the triangle will have legs with length equal to 1 and the hypotenuse will have length !2. Notice that these lengths satisfy the Pythagorean

theorem: 12 1 12 5 A !2 B2, or 2 5 2. Later when we discuss the unit circle approach, we

!2 !2

will let the hypotenuse have length 1. The legs will then have lengths and :

2

2

a !2 b 2 1 a !2 b 2 5 1 1 1 5 12

2

2

22

EXAMPLE 5 Solving a 458-458-908 Triangle

A house has a roof with a 45? pitch (the angle the roof

makes with the house). If the house is 60 feet wide,

x

x

what are the lengths of the sides of the roof that form

the attic? Round to the nearest foot.

45?

45?

60 ft

Solution: Draw the 45?-45?-908 triangle.

Let x represent the length of the unknown legs.

x

x

45?

45?

60 ft

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