# Length of hypotenuse using one side length and angle

I bet this question has been asked a million times, but I can't find a straight answer. I need to find the length of the hypotenuse in a triangle where I have one side and all the angles.

Example:

Now in the above triangle I have the length of a = 20 and all the angles. How do I - from here - get the length of the hypotenuse (c)?

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according to the formula $c\cdot\cos 30 = a$ – W_D Sep 4 '13 at 12:41

just use Law of sines (https://en.wikipedia.org/wiki/Law_of_sines): it states that

$$\frac{a}{\sin{\alpha}} = \frac{b}{\sin{\beta}} = \frac{c}{\sin{\gamma}}$$

where $\alpha, \beta, \gamma$ are the angles opposited to sides $a, b, c$ respectively. Since $\gamma$ is a right angle, $\sin{\gamma} = 1$, and therefore in your example $c = \frac{a}{\sin{60°}}$.

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Thanks a lot. Exactly what I needed. Who knew it would be so simple :) – Trenskow Sep 4 '13 at 13:05
This is shooting an ant with a cannon! Just use the right-angle definitions of $\sin$ and $\cos$, as an earlier comment suggested. $a/c = \text{opposite/hypotenuse}=\sin 60^\circ$. – Ted Shifrin Sep 4 '13 at 15:02
if you have a nail to pe put on a wall, everything is a hammer :-) – mau Sep 5 '13 at 7:04

Maybe for that problem the short path is seeing the triangle as the half of an equilateral triangle. Therefore $20$ is the height and you know the relationship between the size and the height:

$$h=s\frac{\sqrt{3}}{2} \implies s=\frac{2h}{\sqrt{3}}=\frac{40}{\sqrt{3}}$$

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