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How does one go about finding the slopes of the lines making a given angle with a line of a given slope ? Say, we have a line of slope m and are said to find the slopes of lines making angle of $\theta$ with said line. How does one find the slopes?

PS: It is obvious that there must be two lines. But my question is how can one find out both slopes from a single concise formula? Derivation of said formula will be appreciated.

SaitamaSensei
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1 Answers1

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You have that $m=\tan \alpha$ where $\alpha$ is the angle that forms the line with the $OX$-axis. Now, if we have a line that forms an angle $\theta$ with the given line then its slope is $\tan(\alpha+\theta)$ or $\tan(\alpha-\theta).$ In the first case we have:

$$\tan(\alpha+\theta)=\dfrac{\tan\alpha+\tan\theta}{1-\tan \alpha\tan\theta}=\dfrac{m+\tan\theta}{1-m\tan\theta}.$$ In the second case

$$\tan(\alpha-\theta)=\dfrac{\tan\alpha-\tan\theta}{1+\tan \alpha\tan\theta}=\dfrac{m-\tan\theta}{1+m\tan\theta}.$$

Edit

If we call $m'=\tan (\alpha+\theta)$ then we have $$\tan\theta=\dfrac{m'-m}{1+mm'}.$$ If we call $m'=\tan (\alpha-\theta)$ then we have $$\tan\theta=\dfrac{m-m'}{1+mm'}.$$ So, you have, $$\tan\theta=\left|\dfrac{m-m'}{1+mm'}\right|$$ When you solve this you get the values above $$\dfrac{m+\tan\theta}{1-m\tan\theta},\quad\dfrac{m-\tan\theta}{1+m\tan\theta}.$$

mfl
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  • is there a way to unify these two formulas using a modulus symbol? – SaitamaSensei Oct 29 '16 at 19:18
  • i saw a solution which went as follows: The angle between two lines is tan($\theta$)=|(m'-m)/1+m'm| and then it proceeded to solve the equation by solving for m' and getting two distinct answers, with m' standing for slopes of other possible lines. How does this route reconcile with the above explanation? – SaitamaSensei Oct 29 '16 at 19:24
  • I have editted the answer to clarify what you have commented. I hope it's clearer now. – mfl Oct 29 '16 at 19:32
  • Thanks alot! i am slow on the uptake so i'll work on it :) – SaitamaSensei Oct 29 '16 at 19:46
  • the book i used never went about it this way. It just randomly started the solution with using the generalized formula for angle between two lines, the one derived by combining expressions for tan($\theta$) and tan($\pi$-$\theta$) leading me to believe there was a sign convention involved. But now i think the solution was basically a poor approach. – SaitamaSensei Oct 29 '16 at 20:00
  • Actually they are equivalent ways of saying the same. I have given the solutions explicitly. In other case you need to solve for $m'.$ – mfl Oct 29 '16 at 20:08