Download Trigonometry Review - BYU Math Department

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() = adjacent/hypotenuse = 4/5
2) tan () = opposite/adjacent = 3/4
3) csc() = 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:
II. The 45 – 45 – 90 Triangle
The sides of the 45 – 45 – 90 triangle are always in a ratio of 1:1:
:2
Radian Measure
Radian measure, like degree measure, is a way of describing the width of an angle. A circle has 360
degrees or 2 radians. 2 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
There is a simple conversion for
degrees to radians, and vice versa:
Degrees
Radian
1 radian = 180/ degrees
0
0
1 degree = /180 radian
30
/6
45
/4
60
/3
90
/2
180

270
3/2
360
2
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