Transcript 5 6 Inverse Trig 01

5-6: Inverse Trig Functions, 1
Objectives:
Assignment:
1. To define and use
inverse trig functions
• P. 377: 5-27 odd, 31, 33
2. To find derivatives of
inverse trig functions
• P. 378-379: 41-45 odd,
50, 55, 63, 73, 75, 83-89,
94
Warm Up
Evaluate arcsin
1
−
2
.
You will be able to define and
use inverse trig functions
Objective 1
Master of My Domain
In order for the inverse of y = sin x to be a function,
we have to restrict the domain. Refer again to
the graph of f (x) = sin x and find a sensible
interval for which f −1(x) is a function.
Master of My Domain
While any number of intervals would work,
the most convenient is [−π/2, π/2].
Increasing, has a full range of values, contains
all the acute angles of a triangle, 1-1
Inverse of Sine
The inverse sine function is defined by
y  arcsin x if and only if sin y  x
where  1  x  1 and 

2
 y

2
Inverse of Sine
The inverse sine function is defined by
y  arcsin x
Read “the arcsine of x”
Means “the arc whose sine is x”
Inverse of Sine
The inverse sine function is defined by
y  sin 1 x if and only if sin y  x


where  1  x  1 and 
2
 y
2
Inverse of Sine
The inverse sine function is defined by
y  sin 1 x
Domain:  1,1
y  arcsin x
  
Range:   , 
 2 2
Sine ratio
Angle measure
Master of My Domain II
Refer to the graph of f (x) = cos x and find a
sensible interval for which f −1(x) is a
function.
Inverse of Cosine
The inverse cosine function is defined by
y  arccos x if and only if cos y  x
where  1  x  1 and 0  y  
Master of My Domain III
Refer to the graph of f (x) = tan x and find a
sensible interval for which f −1(x) is a
function.
Inverse of Tangent
The inverse tangent function is defined by
y  arctan x if and only if tan y  x


where    x   and 
2
 y
2
Exercise 1
Evaluate.
1. arccos 0
2. arctan 3
3. arcsin 0.3
Interval Property
sin  arcsin x   x cos  arccos x   x tan  arctan x   x
Interval Property
Inverse Properties
arcsin  sin y   y arccos  cos y   y arctan  tan y   y
1  x  1


2
 y

2
1  x  1
0 y 
   x  


2
 y

2
Interval Property
Interval Property
Inverse Properties
csc arccsc 𝑥 = 𝑥 sec arcsec 𝑥 = 𝑥 cot arccot 𝑥 = 𝑥
𝑥 ≥1
𝑥 ≥1
   x  
arccsc csc 𝑦 = 𝑦 arcsec sec 𝑦 = 𝑦 arccot cot 𝑦 = 𝑦
−𝜋/2 ≤ 𝑦 < 0
or 0 < 𝑦 ≤ 𝜋/2
0 ≤ 𝑦 < 𝜋/2 or
𝜋/2 < 𝑦 ≤ 𝜋
0<𝑦<𝜋
Exercise 2
Solve for 𝑥: arctan 2𝑥 − 3 =
𝜋
.
4
More Compositions
To make these compositions a bit more fun, what if
we mismatched the trig functions with their
inverses? To do these problems, draw a right
triangle and use the Pythagorean Theorem.

13
 5 
csc arctan      
5
 12  

tan  
5
12
tan  
5
Quadrant II
12
Quadrant IV
Trig Into Algebra
When you take the derivative of an inverse
trig function, you’ll have to turn that inverse
trig expression into an algebraic one.
2

 x 
9

x
tan arccos    
 3 

x
cos  
x
3
9  x2
Quadrant I
Exercise 3
Evaluate each of the following.
1. cos arcsin 𝑥
2. tan arcsec
5
2
Objective 2
You will be able to
find derivatives of
inverse trig
functions
Exercise 4
Use 𝑓 𝑥 = sin 𝑥 and 𝑓 −1 𝑥 = arcsin 𝑥 to
find the derivative of 𝑓 −1 𝑥 = arcsin 𝑥.
Exercise 5
Find the inverse of 𝑦 = arcsin 2𝑥.
Exercise 6
1
sin 𝑥
2
Use 𝑓 𝑥 =
and 𝑓 −1 𝑥 = arcsin 2𝑥 to
find the derivative of 𝑓 −1 𝑥 = arcsin 2𝑥.
Exercise 7
1. Use 𝑓 𝑥 = cos 𝑥 and 𝑓 −1 𝑥 = arccos 𝑥 to
find the derivative of 𝑓 −1 𝑥 = arccos 𝑥.
2. Use 𝑓 𝑥 = tan 𝑥 and 𝑓 −1 𝑥 = arctan 𝑥 to
find the derivative of 𝑓 −1 𝑥 = arctan 𝑥.
3. Use 𝑓 𝑥 = sec 𝑥 and 𝑓 −1 𝑥 = arcsec 𝑥 to
find the derivative of 𝑓 −1 𝑥 = arcsec 𝑥.
Derivatives of Inverse Trig
Exercise 8
Find each derivative.
1. 𝑦 = arcsin 2𝑥
2. 𝑦 = arctan 3𝑥
3. 𝑦 = arcsin 𝑥
Exercise 9
Find the derivative of 𝑓 𝑥 = arcsec 𝑒 2𝑥 .
Exercise 10
Find the derivative of 𝑦 = arcsin 𝑥 + 𝑥 1 − 𝑥 2 .
Exercise 11
Analyze the graph of
𝑦 = arctan 𝑥 2 .
Exercise 12
A photographer is taking a
picture of a four-foot
painting hung in an art
gallery. The lens is 1 foot
below the lower edge of
the painting. How far
should the camera be
from the painting to
maximize the angle
subtended by the camera
lens?
5-6: Inverse Trig Functions, 1
Objectives:
Assignment:
1. To define and use
inverse trig functions
• P. 377: 5-27 odd, 31, 33
2. To find derivatives of
inverse trig functions
• P. 378-379: 41-45 odd,
50, 55, 63, 73, 75, 83-89,
94