Question

The value of $\cos 66+\sin 84$ is

Answer 1

Determine the value of $\begin{array}{rcl}& & \cos {66}^{\mathrm{o}}+\sin {84}^{\mathrm{o}}\end{array}$.

This value of the expression can be calculated as:

$\begin{array}{rcl}\cos {66}^{\mathrm{o}}+\sin {84}^{\mathrm{o}}& =& \cos {66}^{\mathrm{o}}+\sin {(90-6)}^{\mathrm{o}}\\ & =& \cos {66}^{\mathrm{o}}+\cos 6^{\mathrm{o}}[\because \sin (90-\theta )=\cos \theta ]\\ & =& 2\cos \frac{72}{2}\cos \frac{60}{2}\left[\because \cos \mathrm{A}+\cos \mathrm{B}=2\cos \frac{\mathrm{A}+\mathrm{B}}{2}\cos \frac{\mathrm{A}-\mathrm{B}}{2}\right]\\ & =& 2\cos {36}^{\mathrm{o}}\cos {30}^{\mathrm{o}}\\ & =& 2\times \left(\frac{\sqrt{5}+1}{4}\right)\times \frac{\sqrt{3}}{2}\\ & =& \frac{\sqrt{15}+\sqrt{3}}{4}\end{array}$

Therefore, the required value is $\begin{array}{rcl}& & \cos \end{array}\begin{array}{rcl}& & \end{array}\begin{array}{rcl}& & 66\end{array}\begin{array}{rcl}& & \end{array}$