Find the principal solutions of each of the following equations :

(i)

(ii)

(iii) tan x = -1

(iv)

(v)

(vi)

To Find: Principal solution.

(i) Given:

Formula used: sin = sin = n + (-1)ⁿ , n I

By using above formula, we have

= -sin = sin(= sin x = n +(-1)ⁿ

Put n= 0 x = 0 +(-1)⁰ x =

Put n= 1 x = 1 +(-1)¹ x = ¹ x = =

[ NOTE: = ]

So principal solution is x= and

(ii) Given: cosx =

Formula used: cos = cos = 2n , n I

By using above formula, we have

cosx = = cos x = 2n, n I

Put n= 0 x = 2 × 0 × x =

Put n= 1 x = 2 x = , x = ,

[ 2 So it is not include in principal solution]

So principal solution is x= and

(iii) Given: tan x = -1

Formula used: tan = tan = n , n I

By using above formula, we have

tan x = -1 = tan x = n, n I

Put n= 0 x = n x =

Put n= 1 x = x = x =

So principal solution is x= and

(iv) Given: cosec x =

We know that cosec sin = 1

So sinx =

Formula used: sin = sin = n + (-1)ⁿ , n

By using above formula, we have

sinx = = sin = n +(-1)ⁿ

Put n= 0 x = 0 +(-1)⁰ x =

Put n= 1 x = 1 +(-1)¹ x = ¹ x = =

[ NOTE: = ]

So principal solution is x= and

(v) Given: tan x = -

Formula used: tan = tan = n , n I

By using above formula, we have

tan x = - = tan x = n, n I

Put n= 0 x = n x =

Put n= 1 x = x =

So principal solution is x= and

(vi) Given: sec x =

We know that sec cos = 1

So cosx =

Formula used: cos = cos = 2n , n I

By using the above formula, we have

cosx = = cos x = 2n, n I

Put n= 0 x = 2n x =

Put n= 1 x = 2 x = , x = ,

[ 2 So it is not include in principal solution]

So principal solution is x= and