First slide

Trigonometric equations

Question

$G e n e r a l s o l u t i o n o f (\sqrt{3} - 1) \cos θ + (\sqrt{3} + 1) \sin θ = 2 i s$

Moderate

A

$n π + {(- 1)}^{n} \frac{π}{4} - \frac{π}{12}$
B

$\frac{n π}{2} + \frac{π}{4}$
C

$n π + {(- 1)}^{n} \frac{π}{4} + \frac{π}{12}$
D

$n π + \frac{π}{4}$

Solution

$W e h a v e (\sqrt{3} - 1) \cos θ + (\sqrt{3} + 1) \sin θ = 2 \Rightarrow \sin θ \frac{\sqrt{3} + 1}{2 \sqrt{2}} + \cos θ \frac{\sqrt{3} - 1}{2 \sqrt{2}} = \frac{2}{2 \sqrt{2}} \Rightarrow \sin θ \cos \frac{π}{12} + \cos θ \sin \frac{π}{12} = \frac{1}{\sqrt{2}}$
$\Rightarrow \sin (θ + \frac{π}{12}) = \sin \frac{π}{4} \Rightarrow θ + \frac{π}{12} = n π + {(- 1)}^{n} \frac{π}{4}, n \in Z \Rightarrow θ = n π + {(- 1)}^{n} \frac{π}{4} - \frac{π}{12}, n \in Z$

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