Trigonometry Review

Trigonometry is the study of triangles, particularly right triangles. Trigonometric functions (such as sine and cosine) relate the angles of triangles to the sides of triangles.

Trig Functions

There are six trigonometric functions. Each function represents the ratio of the lengths of two particular sides. For example, the sine of an angle is defined as the ratio of the opposite side to the hypotenuse (the longest side).

The Six Trig Ratios

sin(A) = opposite/hypotenuse (sine)

cos(A) = adjacent/hypotenuse (cosine)

tan(A) = opposite/adjacent (tangent)

csc(A) = hypotenuse/opposite (cosecant)

sec(A) = hypotenuse/adjacent (secant)

cot(A) = adjacent/opposite (cotangent)

You may have noticed that the last three ratios listed are the reciprocals, respectively, of the first three.

Here is a mnemonic that may help you remember which ratio is which:

SOH CAH TOA

Example

Find the cosine, tangent, and cosecant of the indicated angle:

1) cos(q) = adjacent/hypotenuse = 4/5

2) tan (q) = opposite/adjacent = 3/4

3) csc(q) = hypotenuse/opposite = 5/3

The Pythagorean Theorem

To calculate the length of one of a right triangle’s sides when you have the lengths of the other two, use the Pythagorean Theorem:

(a and b are the lengths of the two shorter sides while c is the length of the hypotenuse.)

Special Right Triangles

There are two “special” right triangles, whose sides and angles you might be required to memorize. These triangles are considered special because they have many useful applications.

I. The 30 – 60 – 90 Triangle

The sides of the 30 – 60 – 90 triangle are always in a ratio of 1::2

II. The 45 – 45 – 90 Triangle

The sides of the 45 – 45 – 90 triangle are always in a ratio of 1:1:

Radian Measure

Radian measure, like degree measure, is a way of describing the width of an angle. A circle has 360 degrees or 2p radians. 2p radians in a circle isn’t arbitrary; it’s based on the circumference of a circle with a radius of 1. Radian measure represents the distance you would have to travel along the circumference of a circle with a radius of 1 to reach a particular angle.

Common Angles in Degree and Radian
Degrees / Radian
0 / 0
30 / p/6
45 / p/4
60 / p/3
90 / p/2
180 / p
270 / 3p/2
360 / 2p

The Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin (0,0). Its formula is x2 + y2 = 1.

Table of Common Trig Values

The Graphs of the Six Trig Functions