First slide

Trigonometric equations

Question

If $3 \tan (θ - 15^{\circ}) = \tan (θ + 15^{°})$ , then $θ =$

Moderate

A

$n π + \frac{π}{4}$
B

$n π + \frac{π}{8}$
C

$n π + \frac{π}{3}$
D

none of these

Solution

$W e h a v e \frac{\tan (θ + 15)}{\tan (θ - 15)} = \frac{3}{1}$ ,
$\Rightarrow \frac{\sin (θ + 15) \cos (θ - 15)}{\sin (θ - 15) \cos (θ + 15)} = \frac{3}{1}$
$u \sin g c o m p o n e n d o a n d d i v i d e n d o$
$\Rightarrow \frac{\sin (θ + 15) \cos (θ - 15) + \sin (θ - 15) \cos (θ + 15)}{\sin (θ + 15) \cos (θ - 15) - \sin (θ - 15) \cos (θ + 15)} = \frac{3 + 1}{3 - 1}$
$\Rightarrow \frac{\sin 2 θ}{\sin 30^{0}} = 2 \Rightarrow \sin 2 θ = 1 = \sin \frac{π}{2}$
$\Rightarrow 2 θ = 2 n π + \frac{π}{2} \Rightarrow θ = nπ + \frac{π}{4}$

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