Right-Angled Trigonometry

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

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

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

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

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

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

$\sin 30^{\circ }=\frac{1}{2}$, $\cos 30^{\circ }=\frac{\sqrt{3}}{2}$, $\tan 30^{\circ }=\frac{1}{\sqrt{3}}$.

$\sin 45^{\circ }=\cos 45^{\circ }=\frac{\sqrt{2}}{2}$, $\tan 45^{\circ }=1$.

$\sin 60^{\circ }=\frac{\sqrt{3}}{2}$, $\cos 60^{\circ }=\frac{1}{2}$, $\tan 60^{\circ }=\sqrt{3}$. Also $\sin 0^{\circ }=0$, $\cos 0^{\circ }=1$, $\sin 90^{\circ }=1$, $\cos 90^{\circ }=0$.

The $\text{angle of elevation}$ is measured up from the horizontal; the $\text{angle of depression}$ 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.