Find the principal solutions of each of the following equations:

(i)


(ii)


(iii)


(iv)


(v)


(vi)



To Find: Principal solution.


[NOTE: The solutions of a trigonometry equation for which 0x2 is called principal solution]


(i) Given:


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


By using above formula, we have


= sin x = n +(-1)n


Put n= 0 x = 0 +(-1)0 x =


Put n= 1 x = 1 +(-1)1 x = 1 x = =


So principal solution is x= and


(ii) Given:


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


By using above formula, we have


= cos = 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


(iii) Given:


Formula used: tan = tan = n , n I


By using above formula, we have


= 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:


We know that tan cot = 1


So cotx = tanx =


The formula used: tan = tan = n , n I


By using the above formula, we have


tanx = = tan = n, n I


Put n= 0 x = n x =


Put n= 1 x = x =


So principal solution is x= and


(v) Given: cosec x = 2


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 = 0 +(-1)0 =


Put n= 1 = 1 +(-1)1 = 1 = =


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


1
1