AP Calculus AB

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Date:
Block:
5.6 & 5.7 Inverse Trig Functions
Hopefully, you remember a little bit about inverse trig functions from Pre-Calculus. Let’s see…
None of the trig functions have a direct inverse. Why not?
If none of the trig functions have a direct inverse, how can we work with inverse trig functions?
What is the notation for inverse trig functions?
What does it mean to evaluate an inverse trig function?
Example #1 Evaluate each of the following. No, you may not use calculators!
a) arcsin
1
2
b) arccos 0
c) arctan
3
Example #2 Evaluate each of the following. Yes, you can use a calculator!
a) arcsin (0.3)
b) arctan 7
Example #3 Solve the equation: arctan( 2 x 3)
Example #4 Given y = arcsec
c) arc csc( 3)
4
5
, find tan y . (Hint: use a right triangle  )
2
Derivatives and Integrals of Inverse Trig Functions
Yup, you have to memorize these!
d
arcsin u
dx
u
1
d
arc cot u
dx
u2
Derivatives
If u is a differentiable function of x, then
d
d
u
arctan u
arccos u
dx
dx
1 u2
u
1 u2
d
arc sec u
dx
u
|u|
u2
d
arc csc u
dx
1
u
u2
1
u
|u|
u2
1
Luckily, there are only 3 formulas for the integrals…
Integrals
If u is a differentiable function of x and a > 0, then
du
a2
u2
arcsin
u
a
du
u u
du
C
2
a
2
a
2
1
|u|
arc sec
a
a
u
2
1
u
arctan
a
a
C
C
Questions 1-3 – Find the derivative of each of the following:
1) y cos
1
5x7
2) y sec
1
e3 x
Questions 4-8 – Evaluate each of the following integrals:
4)
1
dt
9 4t 2
3) f ( x) arctan(2 x)
5)
7)
1
e2 x 1
x 2
4 x2
dx
9x3 5
6)
dx (Hint: use long division)
x2 9
dx (Hint: break it up)
8)
x
2
dx
(Hint: complete the square)
4x 7

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