**A ratio is a comparison by division.** You divide one factor
into another to see how they compare.
In sports, ratios are constantly flashed on the TV screen. With baseball, we
are shown a player's batting average, which compares his number of hits to
his number of times at bat.

In basketball, a ratio of shots made to shots taken is called shooting
average.

**The ratio of sides of a right triangle is determined by comparing
the lengths of two sides.**

**Example:**

To introduce ratios of the sides,

let's use the 3-4-5 right triangle again.

With each angle, there are **only** six possible ways that the sides can be divided into each other, so there are **only** six possible ratios for each angle. The six ratios for the reference angle shown (indicated by the shading) are as listed:

is the Greek letter **theta**. It is a variable which is used to represent the degrees of an angle when the degrees are unknown.

In the chart above, is used to represent any angle.

For example, the sine function can be read: The sine of **any angle** is the length of the hypotenuse divided into the length of the opposite side.

**This chart shows the functions and the ratios for the reference angle.** Notice that the ratios are expressed as both fractions and decimals.

The hypotenuse is always the longest side. Notice that when the length of the hypotenuse is the numerator, the ratio is always greater than one. When the length of the hypotenuse is the denominator, the ratio is always less than one.

**Here are the standard abbreviations for the functions.**

**Function practice:** Determine the six functions for each angle of these right triangles. Round off to 4 decimal places.

Remember: There are two reference angles in each right triangle. The right angle is never the reference angle.