Question

The value of $sin{78}^{\circ }-sin{66}^{\circ }-sin{42}^{\circ }+sin{60}^{\circ }$ is

• A. $\frac{1}{2}$
• B. $-\frac{1}{2}$
• C. -1
• D. None of these

Answer 1

The correct option is D

None of these

$sin{78}^{\circ }-sin{66}^{\circ }-sin{42}^{\circ }+sin{60}^{\circ }is=sin{78}^{\circ }-sin{42}^{\circ }-sin{66}^{\circ }+sin{60}^{\circ }=2sin\left(\frac{{78}^{\circ }-{42}^{\circ }}{2}\right)cos\left(\frac{{78}^{\circ }+{42}^{\circ }}{2}\right)-sin{66}^{\circ }+sin{60}^{\circ }[\because cosA+cosB=2cos\left(\frac{A+B}{2}\right)cos\left(\frac{A-B}{2}\right)]=2sin{18}^{\circ }cos{60}^{\circ }-sin{66}^{\circ }+sin{60}^{\circ }=2\times \frac{1}{2}sin{18}^{\circ }-sin{66}^{\circ }+\frac{\sqrt{3}}{2}=sin{18}^{\circ }-sin{66}^{\circ }+\frac{\sqrt{3}}{2}=\frac{\sqrt{5-1}}{4}-0.914+\frac{\sqrt{3}}{2}=0.309-0.914+0.866=0.261$