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!:
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))?
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)
Lesson 1: Evaluating Compositions of Functions Find the exact value, if possible. cos(cos-1 2 2 ) = sin-1(sin π) = cos(cos-1 -1.2) =
Lesson 2: Evaluating a Composite Trig Expression Evaluate. cos[sin-1(-½)] = sin[tan-1(1)] =
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)]
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)
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. 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
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