Inverse Trigonometric Functions

Important Terms, Definitions & Formulae

01. Trigonometric Formulae:

Relation between trigonometric ratios

a) sin

tancos

b)

1tan

cot

c) tan .cot 1

d) cos

cotsin

e)

1cosec

sin

f)

1sec

cos

Trigonometric identities

a) 2 2sin cos 1

b) 2 21 tan sec

c) 2 21 cot cosec

Addition / subtraction formulae & some related results

a) sin sin cos cos sin A B A B A B

b) cos cos cos sin sin A B A B A B

c) 2 2 2 2cos cos cos sin cos sin A B A B A B B A

d) 2 2 2 2sin sin sin sin cos cos A B A B A B B A

e) tan tan

tan1 tan tan

A BA B

A B

f) cot cot 1

cotcot cot

B AA B

B A

Transformation of sums / differences into products & vice-versa

a) sin sin 2sin cos2 2

C DC D

C D

b) sin sin 2cos sin2 2

C DC D

C D

c) cos cos 2cos cos2 2

C DC D

C D

d) cos cos 2sin sin2 2

C D C D

C D

e) sin sin2sin cos A B A BA B

f) 2 sin sincos sin A B A BA B

g) cos cos2cos cos A B A BA B

h) cos cos2sin sin A B A BA B

Multiple angle formulae involving 2A and 3A

a)sin2 2sin cosA A A

b) 2 2

sin 2sin cosA A

A

c) 2 2cos2 cos sin A A A

d) 2 2cos cos sin2 2

A A

A

e) 2cos2 2cos 1 A A

f) 22cos 1 cos2 A A

g) 2cos2 1 2sin A A

h) 22sin 1 cos2 A A

i)2

2 tansin 2

1 tan

AA

A

j)2

2

1 tancos2

1 tan

AA

A

k)2

2 tantan2

1 tan

AA

A

l) 3sin3 3sin 4sin A A A

m) 3cos3 4cos 3cos A A A

n)3

2

3tan tantan3

1 3tan

A AA

A

Relations in Different Measures of Angle

Angle in Radian Measure = Angle in Degree Measure ×180

180Angle in Degree Measure = Angle in Radian Measure ×

( ) l

in radian measurer

Also followings are of importance as well:

o1Right angle 90 o1 = 60 , 1 = 60

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o1 = = 0.01745 radians approximately

180 o1 radian = 57 17 45 or 206265 seconds .

General Solutions a) sin sin ( 1 ) ,nx y x n y where n Z .

b) cos cos 2 ,x y x n y where n Z .

c) tan tan ,x y x n y where n Z .

Relation in Degree & Radian Measures

Angles in Degree

0 30 45 60 90 180 270 360

Angles in Radian

0c 6

c

4

c

3

c

2

c

c

3

2

c

2c

In actual practice, we omit the exponent ‘c’ and instead of writing c we simply write and similarly for others.

Trigonometric Ratio of Standard Angles

Degree /Radian 0 30 45 60 90

T – Ratios

0 6

4

3

2

sin 0 1

2

1

2

3

2 1

cos 1 3

2

1

2

1

2 0

tan 0 1

3 1

3

cosec 2 2 2

3 1

sec 1 2

3 2 2

cot 3 1 1

3 0

Trigonometric Ratios of Allied Angles

Angles

2

2

3

2

3

2

2

OR

2 T- Ratios

sin cos cos sin sin cos cos sin sin

cos sin sin cos

cos sin sin cos cos

tan cot cot tan

tan cot cot

tan tan

cot tan tan cot

cot tan tan

cot cot

sec cosec

cosec

sec

sec cosec

cosec

sec sec

cosec sec sec cosec

cosec

sec sec

cosec

cosec

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02. a) 1 1 11,1sin cosec ,

xx

x b) 1 1 1cosec sin , , 1 1,

xx

x

c) 1 1 11,1cos sec ,

xx

x d) 1 1 1sec cos , , 1 1,

xx

x

e)

1

1

1

1

1π

cot , 0

tan

cot , 0

x

x

x

x

x

f)

1

1

1

1

1π

tan , 0

cot

tan , 0

x

x

x

x

x

03. a) 1 1sin sin , 1,1x x x b) 1 1cos π cos , 1,1 x x x

c) 1 1tan tan , Rx x x d) 1 1 | |cosec cosec , 1 xx x

e) 1 1 | |sec π sec , 1 xx x f) 1 1cot π cot , R x x x

04. a) 1 π π

2 2sin sin , xx x b) 1 0 πcos cos , xx x

c) 1 π π

2 2tan tan , xx x d) 1 π π

, 02 2

cosec cosec , x xx x

e) 1 π0 π,

2sec sec , x xx x f) 1 0 πcot cot , xx x

05. a) 1 1 πsin cos , 1,1

2 x x x

b) 1 1 πtan cot , R

2 x x x

c) 1 1 | | 1 or 1π

cosec sec , 1 . .,2

x xx x i e x

06. a) 1 1 1 2 2sin sin sin 1 1

x y x y y x

b) 1 1 1 2 2cos cos cos 1 1 x y xy x y

c)

1

1 1 1

1

, 1

, 0, 0, 1

, 0, 0, 1

tan1

tan tan π tan1

π tan1

xy

x y xy

x y xy

x y

xy

x yx y

xy

x y

xy

d)

1

1 1 1

1

, 1

, 0, 0, 1

, 0, 0, 1

tan1

tan tan π tan1

π tan1

xy

x y xy

x y xy

x y

xy

x yx y

xy

x y

xy

e) 1 11 1

1tan tan tan tan

x y z xyzz

xy yz zxx y

07. a)2

1 1 22 tan sin , | | 1

1

xx x

x

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b)2

1

2

1 12 tan cos , 0

1

xx x

x

c) 1

2

1 22 tan tan , 1 1

1

xx x

x

08. Principal Value: Numerically smallest angle is known as the principal value.

Finding the principal value: For finding the principal value, following algorithm can be followed–

STEP1– Firstly, draw a trigonometric circle and mark the quadrant in which the angle may lie.

STEP2– Select anticlockwise direction for 1st and 2nd quadrants and clockwise direction for 3rd and 4th quadrants.

STEP3– Find the angles in the first rotation. STEP4– Select the numerically least (magnitude wise) angle among these two values. The angle thus found will be the principal value. STEP5– In case, two angles one with positive sign and the other with the negative sign qualify for the numerically least angle then, it is the convention to select the angle with positive sign as principal value.

The principal value is never numerically greater than .

09. Table demonstrating domains and ranges of Inverse Trigonometric functions:

Discussion about the range of inverse circular functions other than their respective principal value branch

We know that the domain of sine function is the set of real numbers and

range is the closed interval [–1, 1]. If we restrict its domain to 3π π

,2 2

,

π π,

2 2

, π 3π

,2 2

etc. then, it becomes bijective with the range [–1, 1].

So, we can define the inverse of sine function in each of these intervals. Hence, all the intervals of sin–1 function, except principal value branch

(here except of π π

,2 2

for sin–1 function) are known as the range of sin–1

other than its principal value branch. The same discussion can be extended for other inverse circular functions.

Inverse Trigonometric Functions i.e., ( )f x Domain/ Values of x Range/ Values of ( )f x

1sin x [ 1, 1] π π,

2 2

1cos x [ 1, 1] [0, π]

1cosec x R ( 1, 1) π π

, {0}2 2

1sec x R ( 1, 1) π[0, π]

2

1tan x R π π

,2 2

1cot x R (0, π)

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10. To simplify inverse trigonometrical expressions, following substitutions can be considered:

Note the followings and keep them in mind:

The symbol 1sin x is used to denote the smallest angle whether positive or negative, the sine

of this angle will give us x. Similarly 1 1 1 1, , , ,cos x tan x cosec x sec x and 1cot x are defined.

You should note that 1sin x can be written as arcsinx . Similarly other Inverse Trigonometric Functions can also be written as arccosx, arctanx, arcsecx etc.

Also note that 1sin x (and similarly other Inverse Trigonometric Functions) is entirely

different from 1( )sin x . In fact, 1sin x is the measure of an angle in Radians whose sine is x

whereas 1( )sin x is 1

sin x (which is obvious as per the laws of exponents).

Keep in mind that these inverse trigonometric relations are true only in their domains i.e., they are valid only for some values of ‘x’ for which inverse trigonometric functions are well defined!

Expression Substitution

2 2 2 2or a x a x

tanθ or cot θ x a x a

2 2 2 2or a x a x

sin θ or cosθ x a x a

2 2 2 2or x a x a

secθ or cosecθ x a x a

or a x a x

a x a x

cos 2θx a

2 2 2 2

2 2 2 2or

a x a x

a x a x

2 2 cos 2θx a

or x a x

a x x

2 2sin θ or cos θ x a x a

or x a x

a x x

2 2tan θ or cot θ x a x a

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