Right-Angled Trigonometry and 3D Geometry

Right-angled trigonometry is the foundation of the whole geometry and trigonometry unit, and it rests on a single idea: in a right triangle, the ratios of the side lengths depend only on the size of the acute angles, not on how large the triangle is. If we label an acute angle $\theta$, the side opposite it is the opposite, the side next to it (that is not the hypotenuse) is the adjacent, and the longest side facing the right angle is the hypotenuse. The three primary ratios are summarized by the mnemonic SOH-CAH-TOA:

$$\sin\theta=\frac{\text{opp}}{\text{hyp}},\qquad \cos\theta=\frac{\text{adj}}{\text{hyp}},\qquad \tan\theta=\frac{\text{opp}}{\text{adj}}.$$

To find an unknown side, choose the ratio that connects the known angle, the known side, and the side you want. To find an unknown ang