TITLE: Length of hypotenuse using one side length and angle
QUESTION [5 upvotes]: 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)?

REPLY [3 votes]: 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, $c = 
\frac{a}{\cos{30°}}$