# Guide to Game Development/Theory/Mathematics/Trigonometry/The basic identities applied to right angle triangles

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## Opposite, Adjacent and Hypotenuse

When you have a right-angled triangle, and you decide on one of the 2 non-right-angled angles, you can then decide on what the names of the sides are.

Opposite - This is the side of the triangle that is opposite of the angle that you have chosen.
Adjacent - This is the side that touches the angle and that goes towards the right angle.
Hypotenuse - This is the same no matter which angle you chose. It's the longest side of the triangle, which is opposite the right angle.

## Soh Cah Toa

Soh Cah Toa is an acronym to help you remember how you can use your knowledge of opposites, adjacents and hypotenuses to find extra pieces of information on a right-angled triangle. Written out as three equations, it looks like:

${\displaystyle \sin(\theta )={\frac {Opposite}{Hypotenuse}}}$

${\displaystyle \cos(\theta )={\frac {Adjacent}{Hypotenuse}}}$

${\displaystyle \tan(\theta )={\frac {Opposite}{Adjacent}}}$

Using these three equations, if you were given two pieces of information (as long as they're both NOT angles), you can any other piece of information on the triangle. Note: the Greek letter, ${\displaystyle \theta }$ is often used with angles to mean, "an unknown angle."

### Inverse trigonometry

There are three main inverse trigonometric functions: ${\displaystyle \sin ^{-1},\cos ^{-1}}$ and ${\displaystyle \tan ^{-1}}$.

If the trigonometric function with an angle gives you a value, then the inverse trigonometric function of the value, will give you the angle.

 Example questions For the following triangle, find x. Answer: ${\displaystyle \tan(60^{\circ })={\frac {4}{x}}}$ ${\displaystyle x={\frac {4}{\tan(60^{\circ })}}}$ Use your calculator... ${\displaystyle x={\frac {4{\sqrt {3}}}{3}}\approx 2.31}$ For the following triangle, find x. Answer: ${\displaystyle \cos(12^{\circ })={\frac {8}{x}}}$ ${\displaystyle x={\frac {8}{\cos(12^{\circ })}}}$ Use your calculator... ${\displaystyle x\approx 8.18}$ For the following triangle, find x. Answer: ${\displaystyle \sin(30^{\circ })={\frac {8}{x}}}$ ${\displaystyle x={\frac {8}{\sin(30^{\circ })}}}$ You could use calculator, but ${\displaystyle \sin(30^{\circ })={\frac {1}{2}}}$, so you don't need to. ${\displaystyle x=16}$ For the following triangle, find θ. Answer: ${\displaystyle \sin(\theta )={\frac {3}{5}}}$ ${\displaystyle \theta =\sin ^{-1}{\bigg (}{\frac {3}{5}}{\bigg )}}$ Use your calculator... ${\displaystyle \theta \approx 36.9^{\circ }}$ For the following triangle, find θ. Answer: As you've been given three sides, there are three ways to do this. I'm only going to show one way, but they will all get the same answer. ${\displaystyle \tan(\theta )={\frac {12}{5}}}$ ${\displaystyle \theta =\tan ^{-1}{\bigg (}{\frac {12}{5}}{\bigg )}}$ Use your calculator... ${\displaystyle \theta \approx 67.4^{\circ }}$ For the following triangle, find x and θ. Answer: As the angles in a triangle add up to 180, and the to unknown angles are the same, then you can tell that ${\displaystyle \theta =45^{\circ }}$. As for the sides, they are also equal, this means you could use a rearranged version of the Pythagorean theorem (${\displaystyle 2x^{2}=10^{2}}$). But as this is about trigonometry, I'm going to do it that way. I could pick any of sin, cos and tan, also I could pick any one of the two angles. I'll go with cos and the bottom left angle. ${\displaystyle \cos(45^{\circ })={\frac {x}{10}}}$ ${\displaystyle x=10\cos(45^{\circ })}$ You could use your calculator, but you can also get it in surd form without. ${\displaystyle x={\frac {10{\sqrt {2}}}{2}}=5{\sqrt {2}}\approx 7.07}$