Lesson Quiz

Lesson Presentation

Lesson 9.3Evaluate Trigonometric Functions of Any Angle

Warm-Up

Standard Accessed: Students will prove, apply, and model trigonometric functions and ratios.

Warm-Up

sin 𝑨=𝟑√𝟑𝟒𝟑𝟒

2. Evaluate

1. In a right triangle ABC, , and is the length of the hypotenuse. Evaluate , and .

𝟏𝟐

tan 𝑨=𝟑𝟓cos 𝑨=𝟓√𝟑𝟒

𝟑𝟒

3. Evaluate √𝟐4. Evaluate √𝟑

Vocabulary

Vocabulary

Is an angle in standard position whose terminal side lies on an axis. Always a multiple of or .

Essential Understandings

How can you evaluate trig. functions of any angle? If you know the coordinates of a point on the

terminal side of the angle, use the general definitions of the six functions. For a quadrantal angle, use the unit circle. If the reference angle is or , find the function value for the reference angle and adjust the sign as needed for the quadrant of the angle. If you can not use any of these methods to find exact values, use a calculator to find approx. function values.

Evaluate trigonometric functions given a pointEXAMPLE 1Let be a point on the terminal side of an angle in standard position. Evaluate the six trigonometric functions of .

SOLUTION

𝒔𝒊𝒏=−𝟒𝟎𝟒𝟏

1. 2.

𝒄𝒐𝒔=−𝟗𝟒𝟏

3.

𝒕𝒂𝒏=𝟒𝟎𝟗

𝒄𝒔𝒄=−𝟒𝟏𝟒𝟎

4. 5.

𝒔𝒆𝒄=−𝟒𝟏𝟗

6.

𝒄𝒐𝒕=𝟗𝟒𝟎

(−9 ,−40 )−9

−4041

The value of r must always be positive. T/F

Use the Unit CircleEXAMPLE 2Use the unit circle to evaluate the six trigonometric functions of .

SOLUTION

sin 𝜽=𝟎

1. 2.

cos𝜽=𝟏

3.

tan𝜽=𝟎

csc 𝜽=𝑼𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅

4. 5.

sec𝜽=𝟏

6.

cot 𝜽=𝑼𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅

Vocabulary

is read as “theta prime”

Find the reference anglesEXAMPLE 3

SOLUTION

Find the reference angle for the given angle.

𝜽′=𝟓𝝅𝟏𝟐

1.

2.

If is its own reference angle, what do you know about ?

is a 1st quadrant angle.Can a reference angle ever have a negative measure?

No, a reference angle is an acute angle.

Vocabulary

CT

S A

Use reference angles to evaluate functionsEXAMPLE 4

SOLUTION

Evaluate.

√𝟐𝟐

1.

− √𝟑𝟑

2.

What should you do differently when finding the reference angle for an angle measured in radians, rather than degrees?

Use instead of and instead of .

Solve a real – world problemEXAMPLE 5

KIS Dragon Soccer Jake kicks the soccer ball at an initial speed of , projected at an angle of . How far will the ball travel horizontally before hitting the ground? Horizontal distance with same height start and end can be modeled by .

The ball traveled horizontally 57.266ft. Before hitting the ground.

𝑑=462

32sin 2(30 °)

𝑑=66.125 √32

Solve a real – world problem (Using Calculator)EXAMPLE 6

Great Adventure The ferriswheel at Great Adventure in New Jersey has a diameter of 140 feet and its base is 10 feet off the ground. Joseph and Roy board a gondola on the ferriswheel and rotate counterclockwise before the wheel temporarily stops. How high above the ground are Roy and Joseph when the wheel stops?

Roy and Joseph are ft. above the ground when the wheel stops.

, Use definition of sine.

sin 68=𝑦70

+80

64.903+80=144.903

Lesson 9.3 Homework:Practice BPractice C “Honors”