Question

Find the equation of the straight line which passes through the point (1,2) and makes such an angle with the positive direction of x-axis whose sine is $\frac{3}{5}$.

Answer 1

Let $\theta$ be the inclination of the line with the positive x-axis.
Then, we have,

$$\begin{gathered} \sin \theta =\frac{3}{5} \\ \Rightarrow \tan \theta =\frac{\sin \theta }{\sqrt{1-{\sin }^2\theta }}=\frac{{\displaystyle \frac{3}{5}}}{\sqrt{1-{\displaystyle \frac{3^2}{5^2}}}}\frac{3}{\sqrt{5^2-3^2}}=\frac{3}{4} \end{gathered}$$

So, the equation of the line that passes through (1, 2) and has slope $\frac{3}{4}$ is

$$\begin{gathered} y-2=\frac{3}{4}\left(x-1\right) \\ \Rightarrow 3x-4y+5=0 \end{gathered}$$

Hence, the equation of the required line is $3x-4y+5=0$