1. 4.7(b) Notes: The Other Inverse Trig. Functions
Date:
4.7(b) Notes: The Other Inverse Trig. Functions
Lesson Objective: To evaluate and graph the other inverse trig function.
CCSS: F-TF Extend the domain of tri­go­no­me­tric functions using the unit circle.
You will need: unit circle, colored pens

2. Lesson 1: Graphing y = cos-1x.
First, find the Key Points for y = cos x for 0 ≤ x ≤ 2π and use the First 3 Key Points:
|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)=

3. Lesson 1: Graphing y = cos-1x.
5 Key Points:
(x1, y1)= (x2, y2)= (x3, y3)= (x4, y4)= (x5, y5)=
Domain of y = cos x: [0, π], Range: [-1, 1]   

4. Lesson 1: Graphing y = cos-1x.
Reverse the coordinates of y = cos x:
(x1, y1)= (x2, y2)= (x3, y3)= (x4, y4)= (x5, y5)=
y = cos-1x: ( , ) ( , ) ( , )
Domain of y = cos x: [0, π], Range: [-1, 1]
Domain of y = cos-1x: [-1, 1], Range: [0, π]   

5. y = cos-1(x) means cos y = x
Lesson 1: Graphing y = cos-1x.
Inverse Cosine Function, y = cos-1(x) or y = arccos x:
y = cos-1(x) means cos y = x
where -1 ≤ x ≤ 1 and 0 ≤ y ≤ π   
It can be thought of as the angle, or the length of the arc in the interval [0, π] whose cosine is x. 

6. Lesson 2: Graphing y = tan-1x
First, find the key points for y = tan x:
1. Asymp.:
2. x-int.: x = = ; ( , ) 2
3. y = -A: x = = ; ( , )
4. y = A: x = = ; ( , )
Domain: ( , ), Range: ( , )
Next, switch the coordinates to get y = tan-1x:
Asymptotes:
( , )  ( , )
Domain: ( , )
Range: ( , )   

7. Lesson 2: Graphing y = tan-1x
Next, switch the coordinates to get y = tan-1x:
1. Asymp.: 
2. x-int.: x = = ; ( , )  ( , ) 2
3. y = -A: x = = ; ( , )  ( , )
4. y = A: x = = ; ( , )  ( , )
Domain: ( , ), Range: ( , )   

8. Lesson 2: Graphing y = tan-1x
Next, switch the coordinates to get y = tan-1x:
1. Asymp.: 
( , )  ( , )
Domain: ( , ), Range: ( , )   

9. Lesson 2: Graphing y = tan-1x Inverse Tangent Function:
Inverse Tangent Function:
y = tan-1(x) means tan y = x
where -∞ ≤ x ≤ ∞ and -π/2 ≤ y ≤ π/2
It can be thought of as the angle in the interval (-π/2, π/2) whose tangent is x.

10. Lesson 3: Finding the Exact Value of Inverse Cosine and Tangent
Find the exact value of the following: Answer the question – what value of x results in the answer within the inverse interval [0, π] for cosine and (-π/2, π/2) for tangent?

11. Lesson 3:
Find the exact value
of the following:
Answer the ques-
tion – what value
of x results in the
answer within the
inverse interval
[0, π] for cosine?
cos-1(1) = x  cos(x) = 1
cos-1(-½) = x  cos(x) = -½

12. Lesson 3:
Find the exact value
of the following:
Answer the ques-
tion – what value
of x results in the
answer within the
inverse interval
(-π/2, π/2) for tangent?
tan-1(-1) = x  tan(x) = -1
tan-1( ) = x  tan(x) =

13. Lesson 4: Using Your Calculator
Use your calculator to find the values of the following in radians and degrees.
cos-1(-⅓) =
tan-1(-35.85) =
cos -1(9/4) =
tan-1( ) =   

14. 3. Find the exact value of the following.
4.7(b): Do I Get It? Yes or No
1. Graph y = cos-12x
2. Graph y = tan-1(½x)
3. Find the exact value of the following.
a. cos-1( ) b. tan-1( 3 )
4. Use your calculator to find the values of the following.
a. cos-1(⅝) b. tan-1(-9.65)