Trigonometry Refresher

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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Trigonometric Functions Defined

Trigonometric functions are simply the ratios between sides of a right triangle

We can also define these less-used functions:

The Pythagorean Relations

If we use a standard reference triangle with hypotenuse = 1, then we have:

From the Pythagorean Theorem, it is obvious that

sin2A + Cos2A = 1

Dividing this formula by Sin squared and Cos squared, we obtain

1 + Cot2A = Csc2A  and Tan2A + 1 = Sec2A

These are the Pythagorean Relations

Relations Between Functions

We can just as easily define our angles this way

Since B = 90-A, we have:

Trigonometric functions are defined for all angles. If our reference triangle has a hypotenuse of 1, then all possible triangles are radii of a unit circle. The general definition of the trigonometric functions is this:

See if you can reason out why the following are true:

Sin 0 = 0 Sin 90 = 1 Sin 180 = 0 Sin 270 = -1
Cos 0 = 1 Cos 90 = 0 Cos 180 = -1 Cos 270 = 0
Tan 0 = 0 Tan 90 = infinity Tan 180 = 0 Tan 270 = -infinity