4.6(c) Notes: Reciprocal Functions Date: 4.6(c) Notes: Reciprocal Functions   Lesson Objective: To understand the graph of y = csc x and y = sec x. CCSS: F-TF Extend the domain of tri­go­no­me­tric functions using the unit circle. You will need: unit circle

What are inverse func­tions anyway? Lesson 1: What Are Inverse Func­tions Anyway?   What are inverse func­tions anyway?   

Lesson 1: What Are Inverse Func­tions Anyway?   What would the graph of the inverse of sine look like?   

Lesson 1: What Are Inverse Func­tions Anyway? y = sin x   y = sin x   

Lesson 1: What Are Inverse Func­tions Anyway? y = sin x x = sin y   y = sin x x = sin y   

Lesson 1: What Are Inverse Func­tions Anyway? y = sin x x = sin y   y = sin x x = sin y Notice x = sin y is NOT a function except within the restricted domain of -1 ≤ x ≤ 1.

Lesson 1: What Are Inverse Func­tions Anyway?   Inverse Sine Function, y=sin-1(x) or y=arcsin x: y = sin-1(x) means sin y = x where -1 ≤ x ≤ 1 and -π/2 ≤ y ≤ π/2    It can be thought of as the angle or the length of the arc in the interval [-π/2, π/2] whose sine is x. 

Lesson 2: Finding the Exact Value of Inverse Sine   To Find the Exact Value of Inverse: Answer the question – what value of x results in the answer within the interval [-π/2, π/2]?   

Lesson 2: Finding the Exact Value of Inverse Sine   To Find the Exact Value of Inverse: Answer the question – what value of x results in the answer within the interval [-π/2, π/2]? Find the exact value of the following: sin-1(1) = x  sin(x) = 1 sin -1 (- 2 2 ) = x  sin(x) = - 2 2 sin-1(-½) = x  sin(x) = -½

 

Lesson 3: Graphing Inverse Sine How to Graph y = sin-1x:   1. Find the key points for y = sin x from -π/2 ≤ x ≤ π/2 and use middle 3 Key Points of -π ≤ x ≤ π. 2. Reverse the points (make x coordinates y and y coordinates x) and graph.     

Lesson 3: Graph y = sin-1x for -1 ≤ x ≤ 1. Find the key points for y = sin x for -π/2 ≤ x ≤ π/2 (use middle 3 Key Points of -π ≤ x ≤ π) |A|: 2. Period, 2π/B: Interval, Period/4: 4. Phase Shift, C/B: Vertical Shift, D: Max: , Min: 5 Key Points: (x1, y1)= (x2, y2)= (x3, y3)= (x4, y4)= (x5, y5)= Domain of y = sin x: [-π/2, π/2], Range: [-1, 1]     

Lesson 3: Graph y = sin-1x for -1 ≤ x ≤ 1. Reverse the coordinates of y = sin x: (x1, y1)= (x2, y2)= (x3, y3)= (x4, y4)= (x5, y5)=  y = sin-1x: ( , ) ( , ) ( , ) Domain of y = sin x: [-π/2, π/2], Range: [-1, 1] Domain of y = sin-1x: [-1, 1], Range: [-π/2, π/2]     

Lesson 4: Using Your Calculator   Use your calculator to find the values. sin -1(- 2 2 ) sin -1( 3 2 ) sin-1(2)   

4.7(a): Do I Get It? Yes or No   Graph y = sin-1 2x over -π/4 ≤ x ≤ π/4. (Hint: Period = π, so find middle 3 Key Points over the interval -π/2 ≤ x ≤ π/2) Find the exact value of the following: a. sin-1(0) b. sin-1( 2 2 ) c. sin-1(-½) Use a calculator to find the values to four decimal places of the following: a. sin-1(¼) b. sin-1(-0.625) c. sin-1( 5 7 )