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