If $tan^2 \theta = \frac{x}{y}$ how can we construct the angle $\theta$?

Question

If we are given the values of $x$ and $y$ and we know that $\tan^2 \theta = \dfrac{x}{y}$ is it possible for us to construct the angle $\theta$?

Answer 1

Continuing from @egreg ‘s work

$\sin ^2 \theta = \dfrac {x}{x + y}$ and $\cos ^2 \theta = \dfrac {y}{x + y}$

Then, $\dfrac {y}{x + y} - \dfrac {x}{x + y} = \cos ^2 \theta - \sin ^2 \theta $

$∴\dfrac {y - x}{y + x} = \cos 2\theta$

Construct the above angle and then bisect it.

Note that this method can effectively avoid the square root construction.

Answer 2

Set $z=x/y$, for simplicity. Then $$ z=\frac{\sin^2\theta}{\cos^2\theta}=\frac{1}{\cos^2\theta}-1 $$ so $$ \cos^2\theta=\frac{1}{z+1}=\frac{y}{x+y} $$ and $$ \sin^2\theta=1-\frac{1}{z+1}=\frac{z}{z+1}=\frac{x}{x+y} $$ Depending on which quadrant $\theta$ lives in, you can choose the signs in $$ \cos\theta=\pm\sqrt{\frac{y\vphantom{A}}{x+y}}, \quad \sin\theta=\pm\sqrt{\frac{x\vphantom{A}}{x+y}} $$

Answer 3

You can create a right triangle with base =$\sqrt{y}$ and height =$\sqrt{x}$, This will satisfy your condition that $$\tan ^2 \theta =\frac{x}{y}$$ And it can be done via a ruler and compass (I hope you can create a right angle)