Right-Angled Trigonometry

Relative to the angle $\theta$: the $\textbf{hypotenuse}$ is opposite the right angle; the $\textbf{opposite}$ is across from $\theta$; the $\textbf{adjacent}$ is next to $\theta$ (not the hypotenuse).

SOH CAH TOA: $\sin\theta = \dfrac{O}{H}$, $\cos\theta = \dfrac{A}{H}$, $\tan\theta = \dfrac{O}{A}$.

Pick the ratio containing the side you know and the side you want.

To find a side, substitute into the ratio and rearrange: if $\cos 38^{\circ} = \dfrac{A}{15}$, then $A = 15\cos 38^{\circ}$.

If the unknown is on the bottom, rearrange carefully: $\tan 50^{\circ} = \dfrac{8}{A}$ gives $A = \dfrac{8}{\tan 50^{\circ}}$.

To find an angle, use the inverse function: $\theta = \sin^{-1}\left(\dfrac{O}{H}\right)$.

$\sin 30^{\circ} = \tfrac12$, $\cos 30^{\circ} = \tfrac{\sqrt3}{2}$, $\tan 30^{\circ} = \tfrac{1}{\sqrt3}$.

$\sin 45^{\circ} = \cos 45^{\circ} = \tfrac{\sqrt2}{2}$, $\tan 45^{\circ} = 1$.

$\sin 60^{\circ} = \tfrac{\sqrt3}{2}$, $\cos 60^{\circ} = \tfrac12$, $\tan 60^{\circ} = \sqrt3$. Also $\sin 0^{\circ} = 0$, $\cos 0^{\circ} = 1$, $\sin 90^{\circ} = 1$, $\cos 90^{\circ} = 0$.

The $\textbf{angle of elevation}$ is measured up from the horizontal; the $\textbf{angle of depression}$ is measured down from the horizontal. They are equal for two points looking at each other (alternate angles).

In 3D, find a right-angled triangle containing the angle you want, often using Pythagoras first to find a missing side of that triangle.