Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

Mathematics (Maths) Class 12

JEE : Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

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Equations Involving Inverse Trigonometric Functions 

Ex.1 Solve Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

Sol. 

Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

Given Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev ...(1)

or cos–1 x√3 = π/2 – cos–1 x

or cos cos1 x√3 = cos (π/2 – cos–1x)

or x√3 = sin cos–1x    or

Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

or x√3 =  Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

Squaring we get 3x2 = 1 – x2  or 4x2 = 1 = x = ± 1/2

Verification : When x = 1/2

L.H.S. of equation = cos1 ( 3 /2) + cos–1 (1/2) = π/6 + π/3 +π/2 = R.H.S. of equation

When x = –1/2

L.H.S. of equation =  cos–1 (– 3 /2) + cos–1 (–1/2) = π – cos–1 ( 3 /2) + π – cos–1 (1/2)

= π – π/6 + π – π/3 = 3p/2 ≠ R.H.S. of equation

∴ x = 1/2 is the only solution

Ex.2 Solve for x : (tan-1 x)2 + (cot-1 x)2Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

Sol. 

Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

 tan–1 x = - π/4, 3 π/4  =  tan–1 x = – π/4;  x = –1,  Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

 

Ex.3 Determine the integral values of ' k ' for which the system , (arc tan x)2 + (arc cos y)2π2 k and tan -1 x + cos -1 y = π /2 posses solution and find all the solutions. 

Sol.

Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev  = 1 - 2 + 8 k ≥ 0 = k ≥ 1/2 ..(2)

From  (1)  and  (2)    k = 1 

Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

G. Inequations involving inverse trigonometric functions 

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

Sol. 

We have, cos–1 x > sin–1 {for cos–1 x to be real; x E [–1, 1]}

2 cos–1 x > π/2  = cos–1 x > π/4   or  cos (cos–1 x) < cos π/4

Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

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

Sol. 

sin–1(sin 5) > x2 – 4x ⇒ sin–1[sin(5 – 2π)] > x– 4x

⇒ x2 – 4x < 5 – 2π ⇒ x– 4x + (2π – 5) < 0

Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

Summation of Series 

Ex.1 Sum the series 

Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

Sol. 

Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

= 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 ,Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev + ...... to ' n ' terms. Also show that , S  = tan -1 3 .

Sol. 

Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

= tan -1 (n + 2)  –  tan -1 (n)

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

Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

Ex.3 If the sum Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev, find the value of k.

Sol.

Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev

Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev .....

Equations Involving Inverse Trigonometric Functions JEE Notes | EduRev 

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