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The line $y = mx$ bisects the angle between the two lines shown below. Find $m$.

 

 

 Feb 25, 2024
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We have a very interesting property, and very useful property when it comes to slope: \(\tan=\frac{O}{A}\), opposite over adjacent, but when put in the coordinate plane, that is \(\tan=\frac{rise}{run}\), and what else is rise over run? Slope! .

Set the angle formed by y=5x and the x axis as \(\theta\), and the angle formed by y=1/6x and the x axis as \(\phi\).

We want to find \(\tan(\phi+\frac{\theta-\phi}{2})=\tan(\frac{\theta+\phi}{2})\).

Use some of our identities:

With a lot of use of tangent identities, we know \(\tan(\theta)=5, \tan(\phi)=1/6\).

Directly applying our tangent angle identity: \(\tan(\theta+\phi)=\frac{5+\frac{1}{6}}{1-\frac{5}{6}}=31\).

Calculate, \(\tan(\theta+\phi)=\frac{\sin(\theta+\phi)}{1+\cos(\theta+\phi)}\).

Draw a triangle and calculate sin and cos.

\(\sin(\theta+\phi)=\frac{31}{\sqrt{962}}\)\(\cos(\theta+\phi)=\frac{1}{\sqrt{962}}\)

Substituting in, we get \(y=(\frac{\sqrt{962}-1}{31})x\) as our line, so \(\bf{m=\frac{\sqrt{962}-1}{31}}\).\(\)

 Feb 27, 2024

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