SOH-CAH-TOA
SOH-CAH-TOA is a mnemonic for remembering the three basic trigonometric ratios: sine, cosine, and tangent.
Formula
\sin = \frac{O}{H}, \quad \cos = \frac{A}{H}, \quad \tan = \frac{O}{A}
Definition
SOH-CAH-TOA is a mnemonic for the three fundamental right-triangle trig ratios: SOH, Sine = Opposite over Hypotenuse; CAH, Cosine = Adjacent over Hypotenuse; TOA, Tangent = Opposite over Adjacent. These ratios depend only on the angle, not the triangle's size, and in the unit-circle framework they extend to all angles: $\sin(\theta) = y$, $\cos(\theta) = x$, $\tan(\theta) = y/x$ for a point $(x, y)$ on the unit circle at angle $\theta$.
Example
Given a right triangle with $\theta = 35^\circ$ and hypotenuse $15$: opposite $= 15\sin(35^\circ) \approx 8.60$, adjacent $= 15\cos(35^\circ) \approx 12.29$, and $\tan(35^\circ) = 8.60/12.29 \approx 0.700$. These definitions are consistent with the unit circle: the $45$-$45$-$90$ triangle inscribed in the unit circle at $\theta = \pi/4$ has legs of $\sqrt{2}/2$ each, confirming $\sin(\pi/4) = \cos(\pi/4) = \sqrt{2}/2$.
Key Insight
Say it like a word, "SOH-KAH-TOH-AH," a chant many students find easy to remember in an exam. The ratios are dimensionless pure numbers, which is what makes them universal: a triangle with a $35^\circ$ angle has the same ratios whether it is the size of a postage stamp or a mountain. The ratio-based definition works only for $0 < \theta < 90^\circ$; the power of the unit circle extension is that it preserves these ratios for acute angles while naturally defining sin, cos, and tan for all real numbers via coordinates.