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# Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

## JEE : Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

The document Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev is a part of the JEE Course Mathematics (Maths) Class 12.
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Equations Involving Inverse Trigonometric Functions

Ex.1 Solve

Sol.

Given  ...(1)

or cosâ€“1 xâˆš3 = Ï€/2 â€“ cosâ€“1 x

or cos cosâ€“1 xâˆš3 = cos (Ï€/2 â€“ cosâ€“1x)

or xâˆš3 = sin cosâ€“1x    or

or xâˆš3 =

Squaring we get 3x2 = 1 â€“ x2  or 4x2 = 1 = x = Â± 1/2

Verification : When x = 1/2

L.H.S. of equation = cos1 ( 3 /2) + cosâ€“1 (1/2) = Ï€/6 + Ï€/3 +Ï€/2 = R.H.S. of equation

When x = â€“1/2

L.H.S. of equation =  cosâ€“1 (â€“ 3 /2) + cosâ€“1 (â€“1/2) = Ï€ â€“ cosâ€“1 ( 3 /2) + Ï€ â€“ cosâ€“1 (1/2)

= Ï€ â€“ Ï€/6 + Ï€ â€“ Ï€/3 = 3p/2 â‰  R.H.S. of equation

âˆ´ x = 1/2 is the only solution

Ex.2 Solve for x : (tan-1 x)2 + (cot-1 x)2

Sol.

tanâ€“1 x = - Ï€/4, 3 Ï€/4  =  tanâ€“1 x = â€“ Ï€/4;  x = â€“1,

Ex.3 Determine the integral values of ' k ' for which the system , (arc tan x)2 + (arc cos y)2Ï€2 k and tan -1 x + cos -1 y = Ï€ /2 posses solution and find all the solutions.

Sol.

= 1 - 2 + 8 k â‰¥ 0 = k â‰¥ 1/2 ..(2)

From  (1)  and  (2)    k = 1

G. Inequations involving inverse trigonometric functions

Ex.1 Find the interval in which cos-1 x > sin-1 x.

Sol.

We have, cosâ€“1 x > sinâ€“1 {for cosâ€“1 x to be real; x E [â€“1, 1]}

2 cosâ€“1 x > Ï€/2  = cosâ€“1 x > Ï€/4   or  cos (cosâ€“1 x) < cos Ï€/4

Ex.2 Find the solution set of the inequation sin-1(sin 5) > x2 - 4x

Sol.

sinâ€“1(sin 5) > x2 â€“ 4x â‡’ sinâ€“1[sin(5 â€“ 2Ï€)] > xâ€“ 4x

â‡’ x2 â€“ 4x < 5 â€“ 2Ï€ â‡’ xâ€“ 4x + (2Ï€ â€“ 5) < 0

Summation of Series

Ex.1 Sum the series

Sol.

= tan -1 (n + 1) (n + 2) - tan -1 n (n + 1)

Put n  =  1 , 2 , 3 , ........ , n and add, we get Sn  = tan -1 (n + 1) (n + 2)  -  tan -1 2

Ex.2 Sum the series to ' n ' terms , + ...... to ' n ' terms. Also show that , Sâˆž  = tan -1 3 .

Sol.

= tan -1 (n + 2)  â€“  tan -1 (n)

Hence, Sn = tan -1 (n + 2)  +  tan -1 (n + 1) - (tan -1 1  +  tan -1 2)

Ex.3 If the sum , find the value of k.

Sol.

.....

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## Mathematics (Maths) Class 12

209 videos|222 docs|124 tests

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