The Six Trigonometric Functions

In a right triangle, one side — the hypotenuse — always sits opposite the right angle. The other two are named relative to an acute angle $\theta$ : the opposite and the adjacent. From these three sides come the six trigonometric functions.

Sides Naming the three sides

Hypotenuse: opposite the right angle — the longest side.

Opposite: the side facing angle

\theta

Adjacent: the side beside

\theta

(not the hypotenuse).

Ratios The six functions

The three main ratios follow the mnemonic SOH-CAH-TOA:

\sin\theta = \dfrac{\text{opp}}{\text{hyp}}

\cos\theta = \dfrac{\text{adj}}{\text{hyp}}

\tan\theta = \dfrac{\text{opp}}{\text{adj}}

The other three are their reciprocals — just flip the fraction: $\csc\theta = \dfrac{\text{hyp}}{\text{opp}}$ , $\sec\theta = \dfrac{\text{hyp}}{\text{adj}}$ , $\cot\theta = \dfrac{\text{adj}}{\text{opp}}$ .

Note Which angle you choose

"Opposite" and "adjacent" depend on which acute angle you look at — swap the angle and they swap too. Only the hypotenuse stays fixed, because it is always opposite the right angle.

The ratios describe the angle, not the size: enlarge the triangle and every ratio is unchanged. Lengthening the adjacent side makes $\theta$ smaller (toward 0°); shortening it makes $\theta$ larger (toward 90°).

Reference The six functions and their reciprocals

Function	Definition	Reciprocal
sin θ	opp ÷ hyp	csc θ = hyp ÷ opp
cos θ	adj ÷ hyp	sec θ = hyp ÷ adj
tan θ	opp ÷ adj	cot θ = adj ÷ opp

Note At 0° and 90°

As $\theta \to 0°$ , the adjacent side becomes the hypotenuse: $\cos 0° = 1$ and $\sin 0° = 0$ .

As $\theta \to 90°$ , the opposite side becomes the hypotenuse: $\sin 90° = 1$ and $\cos 90° = 0$ .

Summary

The hypotenuse is always opposite the right angle and is the longest side.
The six functions are ratios of sides: sin, cos, tan and their reciprocals csc, sec, cot.
The ratios depend on the angle, not the triangle's size — enlarging it changes nothing.
At 0°: cos = 1, sin = 0. At 90°: sin = 1, cos = 0.