Finding the formula of a line when given a line and angle.

Question

Assume I have a line $y_1 = m_1x + b_1$. And I'm given $\theta$ degrees. How do I find a line $\theta$ degrees counterclockwise from the line.

Answer 1

Let $\alpha$ be the polar angle of your given line: $\tan \alpha=m_1$. Let $\varphi$ be the polar angle of the line you're looking for. Your condition is simply: $\varphi=\alpha+\theta$. Hence the slope of the line will be: $$m_2=\tan(\alpha+\theta)=\frac{\tan\alpha+\tan\theta}{1-\tan\alpha\tan\theta}=\frac{m_1+\tan\theta}{1-m_1\tan\theta} .$$