1. 4.7(c) Notes: Compositions of Functions
Date:
4.7(c) Notes: Compositions of Functions
Lesson Objective: To evaluate and graph the compositions of trig function.
CCSS: F-TF Extend the domain of tri­go­no­me­tric functions using the unit circle.
You will need: unit circle
This is Jeopardy!: 

2. Lesson 1: Evaluating Compositions of Functions
If f(x) = 2x, what is f -1(x)?
What is f(f -1(x))?
What is f -1(f(x))?

3. Lesson 1: Evaluating Compositions of Functions Inverse Properties:
Function within: Only if x falls:
sin(sin-1x) = x [-1, 1]
sin-1(sin x) = x [-π/2, π/2]
cos(cos-1x) = x [-1, 1]
cos-1(cos x) = x [0, π]
tan(tan-1x) = x (-∞, ∞)
tan-1(tan x) = x (-π/2, π/2)

4. Lesson 1: Evaluating Compositions of Functions
Find the exact value, if possible.
cos(cos ) =
sin-1(sin π) =
cos(cos ) =

5. Lesson 2: Evaluating a Composite Trig Expression Evaluate.
cos[sin-1(-½)] =
sin[tan-1(1)] =   

6. Lesson 3: Using a Sketch to Evaluate Composite Trig Expressions
Use a sketch to find the exact values.
sin[tan-1 (¾)]
cot[sin-1 (- 5/13)]

7. Lesson 4: Simplifying Trig Expressions as an Algebraic Expression
Use a right triangle to write each expression as an algebraic expression. Assume that x is posi-tive and that the given inverse trig function is defined for the expression in x.
sin(cos-1 2x)
sec(tan-1 x)

8. 1. Find the exact value, if possible.
4.7(c): Do I Get It? Yes or No
1. Find the exact value, if possible.
a. cos(cos-1 0.6) b. sin-1(sin 3π/2) c. cos(cos-1 1.5)
2. Use a sketch to find the exact values.
a. cos(tan-1 5/12) b. cot[sin-1(-⅓)]
3. Use a right triangle to write each expres­sion as an algebraic expression. Assume that x is positive and that the given inverse trig function is defined for the expression in x.
a. cos(sin-1 x) b. sin(tan-1 x)
4. Find the exact value of cos-1(sin 2π/3).
Answers:
1. a b. - π/2 c. not defined
2. a. 12/13 b. -2 √2
3. a. √(1 – x2) b. x√(x2 + 1)
x2 + 1
4. π/6   

9. 4.7(c): Do I Get It? Yes or No
Answers:
1. a. 0.6 b. - π/2 c. not defined
2. a. 12/13 b. -2 √2
3. a. √(1 – x2) b. x√(x2 + 1)
x2 + 1
4. π/6