Question

If $\sin \varphi =\frac{1}{2}$, find $3\cos \varphi -4{\cos }^3\varphi$.

• A. $1$
• B. $0$
• C. $-1$
• D. none

Answer 1

The correct option is B $0$

$\sin \varphi =\frac{1}{2}=\frac{P}{H}$ (Refer to Image)

$B=\sqrt{2^2-1^2}=\sqrt{3}$

$\cos \varphi =\frac{\sqrt{3}}{2}$

$3\cos \varphi -4co{s}^3\varphi$

$\Rightarrow \cos \varphi (3-4{\cos }^2\varphi )$

$\Rightarrow \frac{\sqrt{3}}{2}(3-4\times \frac{3}{4})$

$=0$