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OBJECTIVES:
Evaluate the inverse trigonometric functions
Evaluate the compositions of trigonometric functions
Inverse Trigonometric Functions
RECALL: for a function to have an inverse function, it must be one-to-one – that is, it must pass the Horizontal Line Test.
So consider the graphs of the six trigonometric functions, will they pass the Horizontal Line Test?
Inverse functions
However, if you restrict the domain of the trig functions, you will have a unique inverse function. But in such a restriction, the range will be unchanged, it will take on the full range of values for the trig function. Therefore, allowing the trig function to be one-to-one.
The INVERSE SINE FUNCTION is defined by
where the domain is and the range is
Inverse Trigonometric functions
1arcsin or sin iff siny x y x y x 1,1 ,
2 2
EX 1: If possible, find the exact valueA)
C)
B)
D)
1arcsin
21 3
sin2
1sin 1 arcsin 2
The INVERSE COSINE FUNCTION is defined by
where the domain is and the range is
The INVERSE TANGENT FUNCTION is defined by
where the domain is and the range is
Inverse Trigonometric functions
1arccos or cos iff cosy x y x y x 1,1 0,
1arctan or tan iff tany x y x y x , ,
2 2
EX 2: If possible, find the exact value A)
C)
B)
D)
arccos1 1 2cos
2
1tan 1 arctan 0
Function Domain Range Quadrant of the Unit
Circle Range Values come
from
I and IV
I and II
I and IV
I and II
I and II
I and IV
Inverse Trigonometric Functions
arcsiny x
arccosy x
arctany x
arccoty x
arcsecy x
arccscy x
1,1
1,1
,
,
, 1 1,
, 1 1,
,2 2
0,
,2 2
0,
0, ,2
y
, , 02 2
y
A)
B)
C)
EX 3: Use a calculator to approximate the value, if possible
cot( 0.3541)arc
arcsin 0.92837781
sec 1.2871684arc
A)
EX4: Find the exact value of the composition function
3sin arctan
2
B)
EX4: Find the exact value of the composition function
5tan arccos
13
C)
EX4: Find the exact value of the composition function
cos cos 0.5arc
D)
EX4: Find the exact value of the composition function
5arccos cos
4