1. 6.8 – Trig Inverses and their graphs

2. Quick Review How do you find inverses of functions?
Are inverses of functions always functions?
How did we test for this?

3. Inverse Trig Functions
Original Function
Inverse
y = sin x
y = sin-1 x
y = arcsin x
y = cos x
y = cos-1 x
y = arccos x
y = tan x
y = tan-1 x
y = arctan x

4. Consider the graph of y = sin x
What is the domain and range of sin x?
What would the graph of y = arcsin x look like?
What is the domain and range of arcsin x?
Domain: all real numbers
Range: [-1, 1]
Domain: [-1, 1]
Range: all real numbers

5. Is the inverse of sin x a function?
This will also be true for cosine and tangent.
Therefore all of the domains are restricted in order for the inverses to be functions.

6. Original Functions with Restricted Domain
How do you know if the domain is restricted for the original functions?
Capital letters are used to distinguish when the function’s domain is restricted.
Original Functions with Restricted Domain
Inverse Function
y = Sin x
y = Sin-1 x
y = Arcsin x
y = Cos x
y = Cos-1 x
y = Arccos x
y = Tan x
y = Tan-1 x
y = Arctan x

7. Original Domains  Restricted Domains
Range
y = sin x
all real numbers
y = Sin x
y = cos x
y = Cos x
y = tan x
all real numbers except n,
where n is an odd integer
y = Tan x

8. Complete the following table on your own
Function
Domain
Range
y = Sin x
y = Arcsin x
y = Cos x
y = Arccos x
y = Tan x
all real numbers
y = Arctan x

9. Table of Values of Sin x and Arcsin x
y = Sin x X Y
-π/2
-π/6
π/6
π/2
y = Arcsin x X Y
-π/2
-π/6
π/6
π/2
Why are we using these values?

10. Graphs of Sin x and Arcsin x

11. Table of Values of Cos x and Arccos x
y = Cos x X Y
π/3
π/2
2π/3 π
y = Arccos x X Y
π/3
π/2
2π/3 π
Why are we using these values?

12. Graphs of Cos x and Arccos x

13. Table of Values of Tan x and Arctan x
y = Tan x X Y
-π/2
-π/4
π/4
π/2
y = Arctan x X Y
-π/2
-π/4
π/4
π/2
Why are we using these values?

14. Graphs of Tan x and Arctan x

15. Write an equation for the inverse of y = Arctan ½x
Write an equation for the inverse of y = Arctan ½x. Then graph the function and its inverse.
To write the equation:
Exchange x and y
Solve for y
Let’s graph 2Tan x = y first.
Complete the table:
Then graph!
y = Tan x X Y
-π/2
-π/4
π/4
π/2
x = Arctan ½y
Tan x = ½y
2Tan x = y
Now graph the original function, y = Arctan ½x by switching the table you just completed!

16. Write an equation for the inverse of y = Sin(2x)
Write an equation for the inverse of y = Sin(2x). Then graph the function and its inverse.
To write the equation:
Exchange x and y
Solve for y
Let’s graph y = Sin(2x) first.
Why are these x-values used?
y = Sin2x X Y
-π/4
-π/12
π/12
π/4
x = Sin(2y)
Arcsin(x) = 2y
Arcsin(x)/2 = y
Now graph the inverse function, y = Arcsin(x)/2 by switching the table you just completed!

17. Evaluate each expression

18. Evaluate each expression