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,

\[\sin\theta = \frac{3}{5}\]

\[ \Rightarrow \tan\theta = \frac{\sin\theta}{\sqrt{1 - \sin^2 \theta}} = \frac{\frac{3}{5}}{\sqrt{1 - \frac{3^2}{5^2}}}\frac{3}{\sqrt{5^2 - 3^2}} = \frac{3}{4}\]

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

\[y - 2 = \frac{3}{4}\left( x - 1 \right)\]

\[ \Rightarrow 3x - 4y + 5 = 0\] 

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