Download 4.7A and B Inverse Trig Functions

Trig./Pre-Calc
4.7A Inverse Trig. Functions
Inverse Sine:
Definition of the Inverse Trigonometric Functions:
Function
Domain
Range
y = arcsin x if and only if sin y = x
1  x  1

y = arcos x if and only if cos y = x
1  x  1
0 y 
y = arctan x if and only if tan y = x
  x  

Inverse Properties
If  1  x  1 and   2  y   2 , then
sin(arcsin x) = x
and
arcsin(sin y) = y.
If  1  x  1 and 0  y   , then
cos(arcos x) = x
and
arcos(cos y) = y.
If   2  y   2 , then
tan(arctan x) = x
and
arctan(tan y) = y.

2

2
y
y

2

2
Example #1
Evaluate the given expression without the aid of a calculator.
1
2
a)
arcsin
c)
arctan(  3 )
b)
arctan
3
3
d)
arcsin
3
2
Example #2
Use a calculator to approximate the value. (Round your answers to two decimal places.)
a)
cos 1 0.75
b)
arctan  6
c)
arccos 0.51
Example #3
Use the properties of inverse trigonometric functions to evaluate the expression.
a)
sin arcsin 0.7
b)
cosarccos 0.3
c)
arcsin sin 3 
Example #4
Find the exact value of the expression without using a calculator. (Hint: make a sketch of a right
triangle.)
a)
sin(arctan 34 )
b)
cos(arcsin
5
13
)
Example #5
Write an algebraic expression that is equivalent to the expression. (Hint: sketch a right triangle.)
a)
cot(arctan x)
b)
sin(arccos x)
c)
x 

csc arctan

2

4.7A HW: Pg. 327 #2-8 even, 16-24 even, 30-40 even, 48-58 even, 106