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Nov. 12, 2019

Inverse Trigonometric Identities & Principle value of Inverse Trigonometric function:

Inverse Trignometric Identities &Principle value of Inverse Trignometric function

 

Principle Value of an Inverse Circular Function

The value of an inverse circular function which is numerically least and the set of values for which, the domain is uniquely defined is called the principle value of the function.

No.

y

DOMAIN (x)

RANGE (y)

GENERAL VALUE

A)

B)

 

C)

D)

E)

   

 

F)

 
   

 

 

Problem Solving Trick 

If two equal numerical values with opposite signs are obtained, then the one with positive sign is taken as the principal value of the Inverse circular function. 

 

Properties of Inverse Circular Functions

P-1 Principle of Reciprocality 

For x > 0 and provided that the value of x in each case is such that both sides of the equality are meaningful 

a)

b)

c) and further similar cases

P-2 Self adjusting Property

Since, sin-1x means the principle value of the angle whose sine is x, the formula is valid only when and the formula sin is meaningful only when

a) provided

sin if

b) provided

cos if

c) provided

if

P-3 Express one Inverse Circular Function in terms of other 

a)

b)

c)

P-4 a) where,

b) where,

c) where,

d) where,

P-5 a) where,

b) where,

c) where,

P-6 a)

b)

c) (where, )

Reasoning: If xy > 1, then and hence, lies in -π/2 and 0

But since, x=>0 an y>=0 therefore, tan-1x and tan-1y both lie in π/2 and 0

P-7 a)

b)

c)

P-8 a)

b)

c)