Question

Find$\sin \left(\frac{\mathrm{\pi }}{4}\right)=\_\_\_\_\_\_;\cos \left(\frac{8\mathrm{\pi }}{3}\right)=\_\_\_\_\_\_\_$

Short answer

The value of$\sin \left(\frac{\mathrm{\pi }}{4}\right)=\frac{1}{\sqrt{2}};\cos \left(\frac{8\mathrm{\pi }}{3}\right)=-\frac{1}{2}$

Step-by-step solution

Step 1. Given Information

We have to find the value of

$\sin \left(\frac{\mathrm{\pi }}{4}\right)=\_\_\_\_\_\_;\cos \left(\frac{8\mathrm{\pi }}{3}\right)=\_\_\_\_\_\_\_$

Step 2. Firstly finding the value of sinπ4

We can write

$\sin \left(\frac{\mathrm{\pi }}{4}\right)=\sin {45}^o$

The value of

$\sin {45}^o=\frac{1}{\sqrt{2}}$

So $\sin \left(\frac{\mathrm{\pi }}{4}\right)=\frac{1}{\sqrt{2}}$

Step 3. Now finding the value of cos8π3

$$\begin{gathered} \cos \left(\frac{8\mathrm{\pi }}{3}\right)=\cos \left(3\mathrm{\pi }-\frac{\mathrm{\pi }}{3}\right) \\ \cos \left(\frac{8\mathrm{\pi }}{3}\right)=-\cos \left(\frac{\mathrm{\pi }}{3}\right) \\ \cos \left(\frac{8\mathrm{\pi }}{3}\right)=-\frac{1}{2} \end{gathered}$$