Find the principal solutions of each of the following equations:
(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