# Trigonometric Ratios

"Trigon" is Greek for
triangle
, and "metric" is Greek for measurement. The
**
trigonometric ratios
**
are special measurements of a
**
right triangle
**
(a triangle with one
angle
measuring
$90\xb0$
). Remember that the two sides of a right triangle which form the right angle are called the
**
legs
**
, and the third side (opposite the right angle) is called the
hypotenuse
.

There are three basic trigonometric ratios:
**
sine
**
,
**
cosine
**
, and
**
tangent
**
. Given a right triangle, you can find the sine (or cosine, or tangent) of either of the non-
$90\xb0$
angles.

$\begin{array}{l}\text{sine}=\frac{\text{lengthofthelegoppositetotheangle}}{\text{lengthofhypotenuse}}\text{abbreviated"sin"}\\ \text{cosine}=\frac{\text{lengthofthelegadjacenttotheangle}}{\text{lengthofhypotenuse}}\text{abbreviated"cos"}\\ \text{tangent}=\frac{\text{lengthofthelegoppositetotheangle}}{\text{lengthofthelegadjacenttotheangle}}\text{abbreviated"tan"}\end{array}$

**
Example:
**

Write expressions for the sine, cosine, and tangent of $\angle A$ .

*
*

The length of the leg opposite
*
$\angle A$
*
is
$a$
. The length of the leg adjacent to
$\angle A$
is
$b$
, and the length of the hypotenuse is
$c$
.

The sine of the angle is given by the ratio "opposite over hypotenuse." So,

$\mathrm{sin}\angle A=\frac{a}{c}$

The cosine is given by the ratio "adjacent over hypotenuse."

$\mathrm{cos}\angle A=\frac{b}{c}$

The tangent is given by the ratio "opposite over adjacent."

$\mathrm{tan}\angle A=\frac{a}{b}$

Generations of students have used the mnemonic "
**
SOHCAHTOA
**
" to remember which ratio is which. (
**
S
**
ine:
**
O
**
pposite over
**
H
**
ypotenuse,
**
C
**
osine:
**
A
**
djacent over
**
H
**
ypotenuse,
**
T
**
angent:
**
O
**
pposite over
**
A
**
djacent.)

## Other Trigonometric Ratios

The other common trigonometric ratios are:$\begin{array}{l}\text{secant}=\frac{\text{lengthofhypotenuse}}{\text{lengthofthelegadjacenttotheangle}}\text{abbreviated"sec"}\mathrm{sec}\left(x\right)=\frac{1}{\mathrm{cos}\left(x\right)}\\ \text{cosecant}=\frac{\text{lengthofhypotenuse}}{\text{lengthofthelegoppositetotheangle}}\text{abbreviated"csc"}\mathrm{csc}\left(x\right)=\frac{1}{\mathrm{sin}\left(x\right)}\\ \text{secant}=\frac{\text{lengthofthelegadjacenttotheangle}}{\text{lengthofthelegoppositetotheangle}}\text{abbreviated"cot"}\mathrm{cot}\left(x\right)=\frac{1}{\mathrm{tan}\left(x\right)}\end{array}$

**
Example:
**

Write expressions for the secant, cosecant, and cotangent of $\angle A$ .

*
*

The length of the leg opposite $\angle A$ is $a$ . The length of the leg adjacent to $\angle A$ is $b$ , and the length of the hypotenuse is $c$ .

The secant of the angle is given by the ratio "hypotenuse over adjacent". So,

$\mathrm{sec}\angle A=\frac{c}{b}$

The cosecant is given by the ratio "hypotenuse over opposite".

$\mathrm{csc}\angle A=\frac{c}{a}$

The cotangent is given by the ratio "adjacent over opposite".

$\mathrm{cot}\angle A=\frac{b}{a}$