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)?
 A: 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 $c = 
\frac{a}{\sin{60°}}$.
A: This staple of trigonometry tends to be taught in schools using the mnemonic "SOHCAHTOA":


*

*Sine of the angle = Opposite side length / Hypotenuse length.


*

*$\sin θ = \frac{Opposite}{Hypotenuse}$


*Cosine of the angle = Adjacent side length / Hypotenuse length


*

*$\cos θ = \frac{Adjacent}{Hypotenuse}$


*Tangent of the angle = Opposite side length / Adjacent side length


*

*$\tan θ = \frac{Opposite}{Adjacent}$



"Adjacent" meaning the side which the given angle joins to the hypotenuse (a if you use the angle 30°), and "Opposite" meaning the side that is not connected to the angle you are using for the calculation (a if you use the angle 60°).
Since both angles are available (even if 60° wasn't stated, you could calculate it as 180 - 90 - 30), you can choose which of the first two to use, flipping the formula to either of these (both are equivalent):


*

*Hypotenuse length = Opposite side length / Sine of the angle


*

*$Hypotenuse = \frac{Opposite}{\sin θ}$ or in example, $c = \frac{a}{\sin{60°}}$


*Hypotenuse length = Adjacent side length / Cosine of the angle


*

*$Hypotenuse = \frac{Adjacent}{\cos θ}$ or in example, $c = 
\frac{a}{\cos{30°}}$


A: 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:
$$h=s\frac{\sqrt{3}}{2} \implies s=\frac{2h}{\sqrt{3}}=\frac{40}{\sqrt{3}}$$
A: When it comes to 30 60 90 triangles, the short leg equals half of the hypotenuse, and the long leg equals the short times the square root of three. 
