If 2 =3 4 , then ^3 + ^3 = - Vidyadip

MCQ

If $\sin 2 \theta=\frac{3}{4}$, then $\sin ^3 \theta+\cos ^3 \theta=$

A
$\frac{\sqrt{5}}{8}$
B
$\frac{\sqrt{7}}{8}$
C
$\frac{\sqrt{11}}{8}$
✓
$\frac{5 \sqrt{7}}{16}$

✓

Answer

Correct option: D.

$\frac{5 \sqrt{7}}{16}$

(D)
$\sin ^3 \theta+\cos ^3 \theta$
$=(\sin \theta+\cos \theta)\left(\cos ^2 \theta+\sin ^2 \theta-\frac{\sin 2 \theta}{2}\right)$
$=\sqrt{(\sin \theta+\cos \theta)^2}\left(1-\frac{\sin 2 \theta}{2}\right)$
$\Rightarrow \sin ^3 \theta+\cos ^3 \theta=\sqrt{1+\frac{3}{4}}\left(1-\frac{3}{8}\right)$
$=\frac{\sqrt{7}}{2} \times \frac{5}{8}=\frac{5 \sqrt{7}}{16}$

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