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Question
If `sqrt3 cos x − sin x = 1`, then general value of x is ______.
Options
`2npi +- pi/3`
`2npi +- pi/6`
`2npi +- pi/3 - pi/6`
`npi + (- 1)^n pi/3`
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Solution
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`
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