Finding slope of a line given angle it makes with a line of a given slope

Question

Given the slope of a line and the angle it makes with another line, how to find the slope of the other line?

Answer 1

That is pretty easy. Let $\theta_1$ and $m_1$ be the angle and slope if the first line, respectively, $\theta_2$ and $m_2$ be the angle and slope if the second line, respectively, and $\phi$ the angle between $\theta_1$ and $\theta_2$.

Then $\theta_1 = \tan^{-1} m_1$ by the definition of tan.

$$\theta_2=\theta_1+\phi$$.

Since $$m_2=\tan \theta_2$$

Finally, we get:

$$m_2=\tan (\phi +(\tan^{-1} m_1))$$.