Question

The value of $si{n}^3{10}^{\circ }+si{n}^3{50}^{\circ }-si{n}^3{70}^{\circ }$ is equal to

• A. $-\frac{3}{2}$
• B. $\frac{3}{4}$
• C. $-\frac{3}{4}$
• D. $-\frac{3}{8}$

Answer 1

The correct option is D

$-\frac{3}{8}$

We have $si{n}^3{10}^{\circ }+si{n}^3{50}^{\circ }-si{n}^3{70}^{\circ }$

$=\frac{1}{4}\left[\right(3sin{10}^{\circ }-sin{30}^{\circ })+(3sin{50}^{\circ }-sin{150}^{\circ })-(3sin{70}^{\circ }-sin{210}^{\circ }\left)\right]$

$=\frac{1}{4}\left[3\right(sin{10}^{\circ }+sin{50}^{\circ }-sin{70}^{\circ })-\frac{3}{2}]$

$=\frac{1}{4}\left[3\right(sin{10}^{\circ }-2cos{60}^{\circ }.sin{10}^{\circ })-\frac{3}{2}]=-\frac{3}{8}$