given y = 5x and y = .5x, When the these lines cross, what is the value of tan theta?
Assuming $\displaystyle \theta$ is the first-quadrant angle between the two lines, suppose that $\displaystyle \alpha$ is the first-quadrant angle between the x-axis and the line y = 0.5. Then,
$\displaystyle \tan \alpha = 0.5$, and
$\displaystyle \tan (\alpha + \theta) = 5$.
Also, $\displaystyle \tan (\alpha + \theta) = \frac{\tan \alpha + \tan \theta}{1 - \tan \alpha \tan \theta} = 5$.
Replace $\displaystyle \tan \alpha$ with 0.5, then solve for $\displaystyle \tan \theta$.