Question: "General solution of (3−1)sin⁡θ+(3+1)cos⁡θ=2"

Lesnaoq73

Answered question

2022-03-25

General solution of

(\sqrt{3} - 1) \sin θ + (\sqrt{3} + 1) \cos θ = 2

Answer & Explanation

Declan Cameron

Beginner2022-03-26Added 12 answers

$(\sqrt{3} - 1) \sin θ + (\sqrt{3} + 1) \cos θ = 2$
$(\frac{\sqrt{3}}{2} - \frac{1}{2}) \sin θ + (\frac{\sqrt{3}}{2} + \frac{1}{2}) \cos θ = 1$
$(\cos 30 - \sin (30)) \sin θ + (\cos 30 + \sin (30)) \cos θ = 1$
$\sin θ \cos 30 - \sin θ \sin (30) + \cos θ \cos 30 + \cos θ \sin (30) = 1$
$\sin (θ + 30) + \cos (θ + 30) = 1$
$\sin (θ + \frac{π}{6}) + \cos (θ + \frac{π}{6}) = 1$
$Clearly the solution set includes \frac{π}{3} + 2 π k and - \frac{π}{6} + 2 π k$
$The trick now to simplify is to take the average value \frac{\frac{π}{3} + - \frac{π}{6}}{2} = \frac{π}{12}$
$The two solutions show up every 2 π k$
$The distance for either solution from \frac{π}{12} is \frac{π}{4}$
$θ = \frac{π}{12} + 2 π k \pm \frac{π}{4}$

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