Trigonometry
High-Yield Summary
- SOH CAH TOA: sinθ = opposite/hypotenuse, cosθ = adjacent/hypotenuse, tanθ = opposite/adjacent — all relative to a chosen angle θ in a right triangle.
- Inverse trig functions (arcsin, arccos, arctan) recover the angle from a ratio; not calculable by hand on the MCAT, but appear conceptually in vector-direction questions.
- Memorize sin/cos/tan at 0°, 30°, 45°, 60°, 90° — these recur constantly across physics problems.
- 30-60-90 triangle (half an equilateral triangle): side ratio 1 : √3 : 2. 45-45-90 triangle (half a square): side ratio 1 : 1 : √2.
SOH CAH TOA
sinθ = opp/hyp, cosθ = adj/hyp, tanθ = opp/adj
- hyp = Hypotenuse — longest side, opposite the 90° angle
- opp = Side opposite angle θ
- adj = Side adjacent to angle θ (not the hypotenuse)
Standard Trig Values
| θ | sinθ / cosθ / tanθ |
|---|---|
| 0° | 0 / 1 / 0 |
| 30° | 1/2 / √3/2 / √3/3 |
| 45° | √2/2 / √2/2 / 1 |
| 60° | √3/2 / 1/2 / √3 |
| 90° | 1 / 0 / undefined |
Special Right Triangles
| 30-60-90 Triangle | 45-45-90 Triangle |
|---|---|
| From bisecting an equilateral triangle | From bisecting a square diagonally |
| Side ratio 1 : √3 : 2 | Side ratio 1 : 1 : √2 |
| Short leg (opp 30°) = 1, long leg (opp 60°) = √3, hyp = 2 | Both legs = 1, hyp (diagonal) = √2 |
Common MCAT Trap
- tan(90°) is undefined, not zero or infinity in a usable sense — watch for this in questions involving angles approaching 90°.
- Don't mix up which leg is 'opposite' vs. 'adjacent' — it depends entirely on which angle (θ) you're referencing, not a fixed side of the triangle.
Quick Recall
A right triangle has a 30° angle and hypotenuse of 10. What is the length of the side opposite the 30° angle?
What is cos(60°)?