Q. 7 - Precalculus Enhanced with Graphing Utilities

Question

 find the exact value of the expression

sin$\left(\frac{\mathrm{\pi }}{6}\right)$

Step-by-Step Solution

Answer

$\sin \left(\frac{\mathrm{\pi }}{6}\right)=\sin 30°=\frac{1}{2}$

Step 1. Given information

sin$\left(\frac{\mathrm{\pi }}{6}\right)$

Step 2. Considering a right angled triangle

Firstly we consider a right angled triangle in which one angle is 30° $\left(\frac{\mathrm{\pi }}{6}\right)$ and the third angle is  $60°\left(\frac{\mathrm{\pi }}{3}\right)$

Let the sides of the triangle be a=1 ,  c=2

So , by Pythagorean theorem

$$\begin{gathered} c^2=a^2+b^2 \\ {b}^2=c^2-a^2 \\ {b}^2=2^2-1^2 \\ b=\sqrt{3} \end{gathered}$$

Step 3. Final value

$$\begin{gathered} \sin \left(\frac{\mathrm{\pi }}{6}\right)=\sin 30°=\frac{a}{c}=\frac{1}{2} \\ so the value of \sin \left(\frac{\mathrm{\pi }}{6}\right) is \frac{1}{2} \end{gathered}$$