This is Maths inverse trigonometry project.

1. Inverse Trigonometric
Functions
Std. XII Maths

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Inverse Sine Function
y

 
2

1

1
x
y = sin x
Sin x has an inverse
function on this interval.
Recall that for a function to have an inverse, it must be a
one-to-one function and pass the Horizontal Line Test.
f(x) = sin x does not pass the Horizontal Line Test
and must be restricted to find its inverse.

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The inverse sine function is defined by
y = arcsin x if and only if sin y = x.
Angle whose sine is x
The domain of y = arcsin x is [–1, 1].
Example:
1
a. arcsin
2 6

 1
is the angle whose sine is .
6 2

1 3
b. sin
2 3


 3
sin
3 2
 
This is another way to write arcsin x.
The range of y = arcsin x is [–/2 , /2].

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Inverse Cosine Function
Cos x has an inverse
function on this interval.
f(x) = cos x must be restricted to find its inverse.
y

 
2

1

1
x
y = cos x

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The inverse cosine function is defined by
y = arccos x if and only if cos y = x.
Angle whose cosine is x
The domain of y = arccos x is [–1, 1].
Example:
1
a.) arccos
2 3

 1
is the angle whose cosine is .
3 2

1 3 5
b.) cos
2 6

  
 
 
 
3
5
cos
6 2
  
This is another way to write arccos x.
The range of y = arccos x is [0 , ].

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Inverse Tangent Function
f(x) = tan x must be restricted to find its inverse.
Tan x has an inverse
function on this interval.
y
x
2
3

2
3
2

2


y = tan x

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The inverse tangent function is defined by
y = arctan x if and only if tan y = x.
Angle whose tangent is x
Example:
3
a.) arctan
3 6

 3
is the angle whose tangent is .
6 3

1
b.) tan 3
3


 tan 3
3
 
This is another way to write arctan x.
The domain of y = arctan x is .
( , )
 
The range of y = arctan x is [–/2 , /2].

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Graphing Utility: Graph the following inverse functions.
a. y = arcsin x
b. y = arccos x
c. y = arctan x
–1.5 1.5
–

–1.5 1.5
2
–
–3 3

–
Set calculator to radian mode.

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Graphing Utility: Approximate the value of each expression.
a. cos–1 0.75 b. arcsin 0.19
c. arctan 1.32 d. arcsin 2.5
Set calculator to radian mode.

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Composition of Functions:
f(f –1(x)) = x and (f –1(f(x)) = x.
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(arccos x) = x and arccos(cos y) = y.
If x is a real number and –/2 < y < /2, then
tan(arctan x) = x and arctan(tan y) = y.
Example: tan(arctan 4) = 4
Inverse Properties:

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Example:
a. sin–1(sin (–/2)) = –/2
 
1 5
b. sin sin
3

  
 
 
5
3
 does not lie in the range of the arcsine function, –/2  y  /2.
y
x
5
3

3


5 2
3 3
 

  
However, it is coterminal with
which does lie in the range of the arcsine
function.
   
1 1
5
sin sin sin sin
3 3 3
  
 
   
   
   
   

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Example:
 
2
Find the exact value of tan arccos .
3
x
y
3
2
adj
2 2
Let =arccos , thencos .
3 hyp 3
u u  
2 2
3 2 5
 
  opp 5
2
tan arccos tan
3 adj 2
u
  
u

13. EXTRA QUESTIONS
• INVERSE TRIG. QUEST.docx
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