Question

Find the angle made by the line x cos ${30}^{\circ }$ + y sin ${30}^{\circ }$ + sin ${120}^{\circ }$ = 0 with the positive direction of the x-axis

Answer 1

Let the required angle be $\theta$. Now, x cos ${30}^{\circ }$ + y sin ${30}^{\circ }$ + sin ${120}^{\circ }$ = 0

$\Rightarrow \left(\frac{\sqrt{3}}{2}\right)x+\left(\frac{1}{2}\right)y+\frac{\sqrt{3}}{2}=0$

$\Rightarrow \sqrt{3}x+y+\sqrt{3}=0$

$\Rightarrow y=(-\sqrt{3})x+(-\sqrt{3})$

$\Rightarrow y=mx+c$, where $m=-\sqrt{3},c=-\sqrt{3}$.

Now, $m=-\sqrt{3}$

$\iff tan\theta =-\sqrt{3}=-tan{60}^{\circ }=tan({180}^{\circ }-{60}^{\circ })=tan{120}^{\circ }$

$\therefore \theta ={120}^{\circ }$

Hence, the given line makes an angle of ${120}^{\circ }$ with the positive direction of the x-axis.