Question

If $sin\, \theta = sin \,15^{\circ}+ sin\, 45^{\circ}$, where $0^{\circ} < \theta < 180^{\circ}$, then $\theta$ is equal to

• A. $45^{\circ}$
• B. $150^{\circ}$
• C. $60^{\circ}$
• D. $75^{\circ}$

Answer 1

Answer: (D)

We have, $sin\,\theta=sin\,15^{\circ}+sin\,45^{\circ}$
$sin\,\theta=2\,sin\left(\frac{15+45}{2}\right)^{\circ}\,cos\left(\frac{15-45}{2}\right)^{\circ}$
$=2\,sin30^{\circ}\,cos\,15^{\circ}$
$=2\times\frac{1}{2}\,cos\,15^{\circ}=cos\,15^{\circ}$
$\Rightarrow sin\,\theta=sin\left(90-15\right)^{\circ}$
$\Rightarrow \theta=75^{\circ}$