Question

How do you simplify $\sin (\frac{\mathrm{\pi }}{6}+\mathrm{x})$?

Answer 1

Step-1: Define sine function:

The sine function is positive in the 1st quadrant. The value of sin $\frac{\mathrm{\pi }}{6}$ is given as $0.5.$

Find the value by:

• Using Trigonometric Functions
• Using Unit Circle

Step-2: Determine the value of the trigonometric function:

Write the expression :

$\sin (\mathrm{A}\pm \mathrm{B})=\mathrm{sinAcosB}\pm \mathrm{cosAsinB}$

$\sin \left(\frac{\mathrm{\pi }}{6}+\mathrm{x}\right)=\sin \left(\frac{\mathrm{\pi }}{6}\right)\mathrm{cosx}+\cos \left(\frac{\mathrm{\pi }}{6}\right)\mathrm{sinx}$

Use the exact value triangle for this angle :
$\sin \left(\frac{\mathrm{\pi }}{6}\right)=\frac{1}{2}$

$\cos \left(\frac{\mathrm{\pi }}{6}\right)=\frac{\sqrt{3}}{2}$

$\Rightarrow \sin \left(\frac{\mathrm{\pi }}{6}+\mathrm{x}\right)=\frac{1}{2}\mathrm{cosx}+\frac{\sqrt{3}}{2}\mathrm{sinx}$

The expression will be simplified as $\mathbf{sin}\left(\frac{\pi }{6}+x\right)\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{cosx}\mathbf{+}\frac{\sqrt{\mathbf{3}}}{\mathbf{2}}\mathbf{sinx}$.