1. Inverse Trig Functions and DifferentiationBy
Dr. Julia Arnold

2. Since a function must pass the horizontal line test to have an inverse function, the trig functions, being periodic, have to have their domains restricted in order to pass the horizontal line test.
For example: Let’s look at the graph of sin x
on [-2pi, 2 pi].
Flunks the
horizontal
line test.

3. By restricting the domain from [- pi/2, pi/2] we produce a portion of the sine function which will pass the horizontal
line test and go from [-1,1].
The inverse sine function is written as y = arcsin(x)
which means that sin(y)=x.
Thus y is an angle and x is a number.
Y = sinx
Y = arcsin x

4. We must now do this for each of the other trig functions:
Y = cos x on [0, pi]
Range [-1,1]
Y= arccos (x) on [-1,1]
range [0, pi]

5. arctan(x) has a range of (- pi/2, pi/2)
arccot(x) has a range of (0, pi)
arcsec(x) has a range of [0, pi], y pi/2
arccsc(x) has a range of [- pi/2, pi/2], y

6. Evaluate without a calculator:
Step 1: Set equal to x
Step 2: Rewrite as
Step 3: Since the inverse is only defined in quadrants 1 & 4 for sin we are looking for an angle in the 4th quadrant whose value is -1/2.
The value must be - pi/6.

7. Evaluate without a calculator:
Step 1: Set equal to x
Step 2: Rewrite as
Step 3: The inverse is only defined in quadrants 1 & 2 for cos so we are looking for the angle whose value is 0.
The value must be pi/2.

8. Evaluate without a calculator:
Step 3: The inverse is only defined in quadrants 1 & 4 for tan so we are looking for the angle whose tan value is sqr(3).
The tan 60 = sqr(3) or x = pi/3

9. Evaluate with a calculator:
Check the mode setting on your calculator. Radian
should be highlighted.
Press 2nd function sin .3 ) Enter.
The answer is .3046

10. Inverse Properties
If -1 < x < 1 and - pi/2 < y < pi/2 then
sin(arcsin x)=x and arcsin(siny)=y
If - pi/2 < y < pi/2 then
tan(arctan x)=x and arctan(tany)=y
If -1 < x < 1 and 0 < y < pi/2 or pi/2 < y < pi then
sec(arcsec x)=x and arcsec(secy)=y
On the next slide we will see how these properties
are applied

11. Inverse Properties Examples
Solve for x:
If - pi/2 < y < pi/2 then
tan(arctan x)=x and arctan(tany)=y
Thus:

12. Inverse Properties ExamplesIf
where 0 < y < pi/2 find cos y
Solution: For this problem we use the right triangle
Sin(y) = x, thus the opp side must be x
and the hyp must be 1, so sin y = x 1 x
By the pythagorean theorem, this makes
the bottom side
Cos(y) =

13. Inverse Properties ExamplesIf
find tan y
Solution: For this problem we use the right triangle
By the pythagorean theorem, this makes
the opp side 1 2
tan(y) =

14. Derivatives of the Inverse Trig Functions

15. Examples Using the Derivatives of the Inverse Trig Functions
Note: u’ = du/dx
Find the derivative of
arcsin 2x
Let u = 2x
du/dx = 2

16. Examples Using the Derivatives of the Inverse Trig Functions
Note: u’ = du/dx
Find the derivative of
arctan 3x
Let u = 3x
du/dx = 3

17. Examples Using the Derivatives of the Inverse Trig Functions
Find the derivative of
Let

18. Some more examples:
Let
find
Solution: Use the right triangle on the coordinate graph
Now using the triangle we can
find sec x after we find the hyp.

19. Some more examples:
Write the expression in algebraic form
Let
then
Solution: Use the right triangle
Now using the triangle we can
find the hyp. 3x y 1

20. Some more examples:
Find the derivative of:
Let u =

21. Please let us know if this presentation has been beneficial.
Thanks.