Question

What is the value of $\cos 75°$?

Answer 1

Find the value of $\cos 75°$.

$\begin{array}{rcl}\cos 75°& =& \cos \left(45+30\right)°\\ & \Rightarrow & \cos 75°=\cos 45°\cos 30°-\sin 45°\sin 30°\left\{\because \cos \left(A+B\right)=\cos A\cos B-\sin A\sin B\right\}\\ & \Rightarrow & \cos 75°=\frac{1}{\sqrt{2}}\left(\frac{\sqrt{3}}{2}\right)-\frac{1}{\sqrt{2}}\left(\frac{1}{2}\right)\left\{\because \cos 45°=\frac{1}{\sqrt{2}},\cos 30°=\frac{\sqrt{3}}{2},\sin 45°=\frac{1}{\sqrt{2}},\sin 30°=\frac{1}{2}\right\}\\ & \Rightarrow & \cos 75°=\frac{\sqrt{3}}{2\sqrt{2}}-\frac{1}{2\sqrt{2}}\\ & \Rightarrow & \cos 75°=\frac{\sqrt{3}-1}{2\sqrt{2}}\end{array}$

Hence, the value of $\begin{array}{l}\cos 75°\end{array}$ is $\begin{array}{l}\frac{\sqrt{3}-1}{2\sqrt{2}}\end{array}$.