9.3 use trig functions to find the measure of the angle

9.3 Use Trig Functions to Find the

Measure of the Angle

NEW VOCABULARY:

“ : reference angle (start angle)

When finding an angle calculator must be in degrees. Push the mode key and see if the calculator is in degrees. If calculator is in radians, change to degrees.

“ Used to represent the reference angle with 3 trig functions

sin = cos tan =

NEW VOCABULARY: Inverse Function: Used to solve for an

angle given two sides of a triangle• Look for the “sin” key on the calculator – to

use inverse sin, you must push the 2nd key and then “sin”. Lets try. Do you get the following inverse functions in the window???

WHAT IS THE MEASURE OF THE ANGLE “” IN THE TRIANGLE SHOWN• What measurement of angles are known?

• How can we find the other two angles?

• What kind of angles are they?

• Identify each side of the triangle using hypotenuse, opposite, and adjacent

LABILE EACH SIDE OF THE TRIANGLE

Hypotenuse “H”

Opposite “O”

Adjacent “A”

EXAMPLE 1 Standardized Test Practice

SOLUTION

In the right triangle, you are given the lengths of the side adjacent to θ and the hypotenuse, so use the inverse cosine function to solve for θ.

cos θ =adjhyp =

611

cos – 1θ = 611

56.9°

The correct answer is C.ANSWER

You Try: for Examples 1

Find the measure of the angle θ.

1.

SOLUTION

In the right triangle, you are given the lengths of the side adjacent to θ and the hypotenuse. So, use the inverse cosine function to solve for θ.

cos θ =adjhyp

= 49

= 63.6°θ cos–1 49

You Try: for Examples 1

Find the measure of the angle θ.

SOLUTION

In the right triangle, you are given the lengths of the side opposite to θ and the side adjacent. So, use the inverse tan function to solve for θ.

2.

tan θ =oppadj

=108

θ 51.3°= tan–1 108

You Try: for Examples 1

Find the measure of the angle θ.

SOLUTION

In the right triangle, you are given the lengths of the side opposite to θ and the hypotenuse. So, use the inverse sin function to solve for θ.

3.

sin θ =opphyp

= 512

24.6°θ = sin–1 512

EXAMPLE 2 Write and solve a trigonometric equation

Monster Trucks

A monster truck drives off a ramp in order to jump onto a row of cars. The ramp has a height of 8 feet and a horizontal length of 20 feet. What is the angle θ of the ramp?

EXAMPLE 2 Write and solve a trigonometric equation

SOLUTION

STEP 1 Draw: a triangle that represents the ramp.

STEP 2 Write: a trigonometric equation that involves the ratio of the ramp’s height and horizontal length.

tan θ =oppadj =

820

EXAMPLE 2 Write and solve a trigonometric equation

STEP 3 Use: a calculator to find the measure of θ.

tan–1θ = 820

21.8°

The angle of the ramp is about 22°.

ANSWER

You Try for Examples 2

4. WHAT IF? In Example 2, suppose a monster truck drives 26 feet on a ramp before jumping onto a row of cars. If the ramp is 10 feet high, what is the angle θ of the ramp?

SOLUTION

STEP 1 Draw: a triangle that represents the ramp.

STEP 2 Write: a trigonometric equation that involves the ratio of the ramp’s height and horizontal length.

tan θ =oppadj

=1026

You try: for Examples 2

STEP 3 Use: a calculator to find the measure of θ.

22.6°tan–1θ =1026

The angle of the ramp is about 22.6°.

ANSWER