Question

Find the slopes of the lines which make the following angles with the positive direction of x-axis:
(i) $-\frac{\mathrm{\pi }}{4}$

(ii) $\frac{2\mathrm{\pi }}{3}$

(iii) $\frac{3\mathrm{\pi }}{4}$

(iv) $\frac{\mathrm{\pi }}{3}$

Answer 1

(i) $\theta =-\frac{\mathrm{\pi }}{4}$

$$\begin{gathered} \therefore \mathrm{Slope}\mathrm{of}\mathrm{the}\mathrm{line}=m=\tan \theta \\ \Rightarrow \mathrm{Slope}\mathrm{of}\mathrm{the}\mathrm{line}=\tan \left(-\frac{\mathrm{\pi }}{4}\right)=-\tan \left(\frac{\mathrm{\pi }}{4}\right)=-1 \end{gathered}$$

Hence, the slope of the line is $-$1.

(ii) $\theta =\frac{2\mathrm{\pi }}{3}$

$$\begin{gathered} \therefore \mathrm{Slope}\mathrm{of}\mathrm{the}\mathrm{line}=m=\tan \theta \\ \Rightarrow \mathrm{Slope}\mathrm{of}\mathrm{the}\mathrm{line}=\tan \left(\frac{2\mathrm{\pi }}{3}\right)=-\tan \left(\frac{\mathrm{\pi }}{3}\right)=-\sqrt{3} \end{gathered}$$

Hence, the slope of the line is $-\sqrt{3}$.

(iii) $\theta =\frac{3\mathrm{\pi }}{4}$

$$\begin{gathered} \therefore \mathrm{Slope}\mathrm{of}\mathrm{the}\mathrm{line}=m=\tan \theta \\ \Rightarrow \mathrm{Slope}\mathrm{of}\mathrm{the}\mathrm{line}=\tan \left(\frac{3\mathrm{\pi }}{4}\right)=-\tan \left(\frac{\mathrm{\pi }}{4}\right)=-1 \end{gathered}$$

Hence, the slope of the line is $-1$.

(iv) $\theta =\frac{\mathrm{\pi }}{3}$

$$\begin{gathered} \therefore \mathrm{Slope}\mathrm{of}\mathrm{the}\mathrm{line}=m=\tan \theta \\ \Rightarrow \mathrm{Slope}\mathrm{of}\mathrm{the}\mathrm{line}=\tan \left(\frac{\mathrm{\pi }}{3}\right)=\sqrt{3} \end{gathered}$$

Hence, the slope of the line is $\sqrt{3}$.