Download The Inverse Trigonometric Functions

Pre-Calculus
Notes
Name: ____________
Date: ____________
Lesson 4.7: The Inverse Trigonometric Functions
Learning Targets:
J: Evaluate the inverse trigonometric functions.
K: Evaluate the composition of trigonometric functions.
N: Use the properties of the inverse trigonometric functions.
Q: Use inverse trigonometric functions to model and solve real-world problems.
X: Graph or identify graphs of the inverse trigonometric functions.
Vocabulary:
inverse sine
inverse cosine
inverse tangent
The Inverse Sine Function:
On the grid below, graph 2 cycles of the function y = sin x.
Find the equation of the inverse of y = sin x. ____________________________
Is the inverse a function? Why or why not?
How could the domain of the sine function, y = sin x, be restricted so that
its inverse was a function?
Things to consider: The restricted domain should include angles between
0 and  , measures of acute angles of right triangle.
2
Restricted Domain
Range should take on all possible values of function.
________________
The function is continuous over restricted domain.
Sine Function
Restricted
Domain: ____________
Range:
____________
Inverse Sine Function
Domain: ____________
Range:
____________
The Inverse Cosine Function:
On the grid below, graph 2 cycles of the function y = cos x.
Find the equation of the inverse of y = cos x. ____________________________
Is the inverse a function? Why or why not?
Cosine Function
Restricted
Domain: ____________
Range:
____________
Inverse Cosine Function
Domain: ____________
Range:
____________
The Inverse Tangent Function:
On the grid below, graph 2 cycles of the function y = tan x.
Find the equation of the inverse of y = tan x. ____________________________
Is the inverse a function? Why or why not?
Tangent Function
Restricted
Domain: ____________
Range:
____________
Inverse Tangent Function
Domain: ____________
Range:
____________
Examples:
1.
a. Evaluate sin 1 ( -
3
), giving an answer in radians.
2
b. Evaluate cos 1 (-1), giving an answer in degrees.
c. Evaluate tan 1 (
2.
3
), giving an answer in radians.
3
a. Evaluate Arcsin(1), giving an answer in degrees.
1
b. Evaluate Arccos   , giving an answer in radians.
2
c. Evaluate Arctan(30), giving an answer in degrees.
3.
A flagpole is 60 feet high. If you sight the top of the pole from ground level,
express the angle of elevation of the top of the flagpole as a function of your
distance d from the top.
4.
Give the angle  the line y = mx makes with the positive part of the x-axis as a
function of the slope of the line.
Composition of Inverse Trigonometric Functions
5.

If 1  x  1 and 

If 1  x  1 and

If x is real # and 

2
y

2
then sin(arcsin x )  x and arcsin(sin y )  y .
0  y   then cos(arccos x)  x and arccos(cos y)  y .

2
y

2
then tan(arctan x)  x and arctan(tan y)  y .
 
a. Evaluate sin 1  sin  , giving an answer in radians.
4

7 

b. Evaluate arccos  cos
 , giving an answer in radians.
6 

c. Evaluate tan 1 (tan 135  ), giving an answer in degrees.
6.
7.
a. Find the exact value: sin (arccos
4
).
5
b. Find the exact value: cos (arctan
5
).
6
c. Find the exact value: tan(arcsin
8
).
89
Write an algebraic expression that is equivalent to the expression.
a. sin(arctan x)
b. sec[arcsin (x-1)]
c. cot(arctan
1
)
x