1. 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

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

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

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

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

6. 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.

7. 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. 

8. 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]?   

9. 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 ( ) = x  sin(x) =
sin-1(-½) = x  sin(x) = -½

11. 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.     

12. 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|: Period, 2π/B:
Interval, Period/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]   

13. 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]   

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

15. 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( ) 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( )