find length of opposite when you only have incomplete length of adjacent

Question

This is probably a really stupid question, but say you have the following:

How do you find the length of x given that we only have an incomplete length of the adjacent here?

Answer 1

Let $y$ be the side opposite the right angle. We have $y=360$ because the angle opposite to side of $360$ is also $35$ degrees: $180°-(180°-70°)-35°$. Then $x=y \sin 70°\approx 338.3$

Answer 2

I guess you could also use a system of equations, as an alternative way. Let $a$ equal the unknown segment of the adjacent side. Write an equation relating the opposite and adjacent sides in both the small right triangle and the large right triangle. $$\begin{cases} \tan 35\unicode{xb0} = \frac{x}{360+a} \implies (360+a)\tan 35\unicode{xb0} = x \\ \\ \tan 70\unicode{xb0} = \frac{x}{a} \implies a\tan 70\unicode{xb0} = x \end{cases} $$

Now, set $x = x$.

$$(360+a)\tan 35\unicode{xb0} = a\tan 70\unicode{xb0}$$

$$360\tan 35\unicode{xb0}+a\tan 35\unicode{xb0} = a\tan 70\unicode{xb0}$$

$$a\tan 70\unicode{xb0}-a\tan 35\unicode{xb0} = 360\tan 35\unicode{xb0}$$

$$a(\tan 70\unicode{xb0}-\tan 35\unicode{xb0}) = 360\tan 35\unicode{xb0} \implies a = \frac{360\tan 35\unicode{xb0}}{\tan 70\unicode{xb0}-\tan 35\unicode{xb0}} \approx 123.12725$$

Now, plug in the value of $a$ in either of the two equations to get $x$.

$$a\tan 70\unicode{xb0} = x \implies x = \frac{360\tan 35\unicode{xb0}\tan 70\unicode{xb0}}{\tan 70\unicode{xb0}-\tan 35\unicode{xb0}} \approx 338.28934$$