Question

If `sqrt3 cos x − sin x = 1`, then general value of x is ______.

पर्याय

• `2npi +- pi/3`

• `2npi +- pi/6`

• `2npi +- pi/3 - pi/6`

• `npi + (- 1)^n pi/3`

Answer 1

If `sqrt3cos x − sin x = 1`, then general value of x is `bbunderline(2npi +- pi/3 - pi/6)`.

Explanation:

Divide the entire equation `sqrt3cos x − sin x = 1` by 2,

`sqrt3/2 cos x − 1/2 sin x = 1/2`

Substitute `cos (pi/6) = sqrt3/2 and sin (pi/6) = 1/2`

`cosx cos(pi/6) - sinx sin(pi/6) = 1/2`

Using the identity cos(A + B) = cosA cosB − sinA sin B

`cos(x + pi/6) = 1/2`

The general solution for cos θ = cos α is θ = 2nπ ± α. Since `cos(pi/3) = 1/2`

`x + pi/6 = 2npi ± pi/3`

Subtract `pi/6` from both sides to get your exact statement:

`x = 2npi ± pi/3 - pi/6`