Sin, cos, and tan
What you'll learn
See the three trig ratios as fixed properties of an angle — not formulas to memorize but ratios that come from the triangle's shape.
Trigonometry starts with a single observation: every right triangle with the same angles has the same shape, no matter how large or small. That means the ratio of any two sides depends only on the angle, not on the triangle's size. Those ratios are sin, cos, and tan.
The three ratios
For a right triangle, pick one of the acute angles and call it A. The three sides get names based on their position relative to A:
- Opposite — the side directly across from A
- Adjacent — the side next to A (not the hypotenuse)
- Hypotenuse — always the longest side, opposite the right angle
The three ratios follow directly from those names:
| Ratio | Reads as | Formula |
|---|---|---|
| sin A | "sine of A" | opposite ÷ hypotenuse |
| cos A | "cosine of A" | adjacent ÷ hypotenuse |
| tan A | "tangent of A" | opposite ÷ adjacent |
The mnemonic SOH-CAH-TOA packages all three: Sin = Opposite / Hypotenuse, Cos = Adjacent / Hypotenuse, Tan = Opposite / Adjacent.
Why the ratios are fixed
If you scale the triangle — make it twice as big — both sides in any ratio double. The quotient stays exactly the same. That's why sin 30° is always 0.5, no matter which 30° triangle you draw: the shape locks in the ratio.
The three reciprocals (csc, sec, cot) are just the three ratios flipped: csc = 1/sin, sec = 1/cos, cot = 1/tan. Six names, three ratios.
Check your understanding
Question 1 of 2
In a right triangle with angle A, the opposite side is 6 and the hypotenuse is 10. What is sin A?