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


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