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

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.

S^oh C^ah T^oa

S^oh C^ah T^oa 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:

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

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

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

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, $\theta$ is often used with angles to mean, "an unknown angle."

Inverse trigonometry

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

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: $\tan(60^{\circ })={\frac {4}{x}}$ $x={\frac {4}{\tan(60^{\circ })}}$ Use your calculator... $x={\frac {4{\sqrt {3}}}{3}}\approx 2.31$ For the following triangle, find x. Answer: $\cos(12^{\circ })={\frac {8}{x}}$ $x={\frac {8}{\cos(12^{\circ })}}$ Use your calculator... $x\approx 8.18$ For the following triangle, find x. Answer: $\sin(30^{\circ })={\frac {8}{x}}$ $x={\frac {8}{\sin(30^{\circ })}}$ You could use calculator, but $\sin(30^{\circ })={\frac {1}{2}}$, so you don't need to. $x=16$ For the following triangle, find θ. Answer: $\sin(\theta )={\frac {3}{5}}$ $\theta =\sin ^{-1}{\bigg (}{\frac {3}{5}}{\bigg )}$ Use your calculator... $\theta \approx 36.9^{\circ }$ 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. $\tan(\theta )={\frac {12}{5}}$ $\theta =\tan ^{-1}{\bigg (}{\frac {12}{5}}{\bigg )}$ Use your calculator... $\theta \approx 67.4^{\circ }$ 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 $\theta =45^{\circ }$. As for the sides, they are also equal, this means you could use a rearranged version of the Pythagorean theorem ($2x^{2}=10^{2}$). 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. $\cos(45^{\circ })={\frac {x}{10}}$ $x=10\cos(45^{\circ })$ You could use your calculator, but you can also get it in surd form without. $x={\frac {10{\sqrt {2}}}{2}}=5{\sqrt {2}}\approx 7.07$