Trigonometry is the study of triangles, but it's really the study of circles and waves. It's how your GPS finds you and how music is digitally stored.
SOH CAH TOA
The three magic words for right-angled triangles:
- SOH: Sine = Opposite / Hypotenuse
- CAH: Cosine = Adjacent / Hypotenuse
- TOA: Tangent = Opposite / Adjacent
The key insight: these ratios depend only on the angle, not the size of the triangle. Every right triangle with a 30° angle has an opposite/hypotenuse ratio of exactly 0.5, whether it's drawn on a postage stamp or spans a canyon. That's what makes trig useful — measure an angle and one side, and the ratios unlock everything else.
Worked Example: Measuring a Flagpole Without Climbing It
You stand 20 meters from the base of a flagpole. Looking up at the top, the angle of elevation is 38°. How tall is the pole?
- Draw the triangle. The ground distance (20 m) is adjacent to the angle; the pole's height is opposite it. The hypotenuse is your line of sight.
- Choose the ratio. We have adjacent, we want opposite — that's Tangent (TOA).
- Set up the equation. tan(38°) = height / 20.
- Solve. height = 20 × tan(38°) = 20 × 0.7813 ≈ 15.6 meters.
- Sanity check. A 38° angle is less than 45°, so the height should be less than the 20 m distance (at exactly 45° they'd be equal). 15.6 < 20 ✓.
This one pattern — angle plus one side gives the rest — is surveying, navigation, and architecture in miniature.
The Unit Circle
If you take a circle with radius 1, the x-coordinate is Cosine and the y-coordinate is Sine. This creates the beautiful waves you see in physics.
The unit circle is also how trig escapes the triangle. A right triangle can only hold angles between 0° and 90°, but a point can travel all the way around a circle — which is why sin(150°) or cos(300°) make sense. As the point orbits, its height traces the sine wave and its horizontal position traces the cosine wave: same wave, shifted by 90°.
The Identities You Actually Need
Trigonometry has hundreds of identities, but a first course really runs on three. The Pythagorean identity, sin²θ + cos²θ = 1, comes straight from the unit circle and the Pythagorean theorem — know one of sine or cosine and it hands you the other. The quotient identity, tan θ = sin θ / cos θ, connects all three main ratios. And the complementary angle identity, sin(θ) = cos(90° − θ), explains the "co" in cosine: the cosine of an angle is the sine of its complement. Most exam simplification problems are these three identities wearing disguises.
Common Mistakes to Avoid
- Wrong calculator mode. Computing tan(38) in radian mode gives a wildly wrong number. Degrees for geometry problems, radians for calculus — and check the mode indicator every time.
- Mixing up opposite and adjacent. "Opposite" and "adjacent" are relative to the angle you're using, not fixed sides of the triangle. Switch the angle and the labels switch too. Always mark the angle first, then label.
- Using SOH CAH TOA on non-right triangles. These ratios require a 90° angle. For other triangles you need the Law of Sines or Law of Cosines.
- Misreading inverse notation. sin⁻¹(x) means arcsine — "which angle has this sine?" — not 1/sin(x). That reciprocal is a different function called cosecant.
Frequently Asked Questions
How can sine and cosine apply to angles bigger than 90°?
Through the unit circle definition. The angle measures rotation from the positive x-axis, and sine/cosine are simply the coordinates of the rotated point. At 150°, the point sits in the second quadrant: sin(150°) = 0.5 (still above the axis) while cos(150°) ≈ −0.87 (now left of center).
Why can't tangent handle 90°?
Tangent is sine divided by cosine, and cos(90°) = 0 — division by zero. Geometrically, a "triangle" with two 90° angles is impossible; on the graph, tangent shoots to infinity with a vertical asymptote at every odd multiple of 90°.
What's the point of radians when degrees work fine?
Radians measure angle by arc length on the unit circle, which makes them the "natural" unit: formulas like arc length = rθ, and the calculus fact that the derivative of sin(x) is cos(x), are only true in radians. Degrees are convenient for measuring; radians are convenient for mathematics.