# 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

• Sine = Opposite / Hypotenuse
• Cosine = Adjacent / Hypotenuse
• Tangent = Opposite / Adjacent = Sine / Cosine

We can also define these less-used functions:

• Cosecant (Csc) = Hypotenuse / Opposite = 1 / Sine
• Secant (Sec) = Hypotenuse / Adjacent = 1 / Cosine
• Cotangent (Cot) = Adjacent / Opposite = Cosine / Sine = 1 / Tangent

## 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:

• Sin A = Cos (90-A)
• Cos A = Sin (90-A)
• Tan A = Cot (90-A)
• Cot A = Tan (90-A)

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
 Sin(-A) = -Sin A Cos(-A) = Cos A Tan(-A) = -Tan A Sin (180-A) = Sin A Cos(180-A) = -Cos A Tan(180-A) = -Tan A Sin (180+A) = -Sin A Cos(180+A) = -Cos A Tan(180+A) = Tan A

## Some Useful Approximations

A very useful way of describing angles is in terms of radians. There are 2 pi or 6.2832.. radians in 360 degrees.

From the above definition, it is easy to see that if a circle has a radius = 1, the length of an arc enclosed by an angle is exactly equal to the angle in radians. More generally, if a circle has radius R, the arc length enclosed in an angle Q is

A = RQ

Artillerymen use a system based on radian measure. They divide a circle into 6400 mils. 6400 is not exactly 2000 times pi but is a lot more convenient to use than 6283. At a distance of 1000 meters, one mil equals very nearly one meter (98.2 cm, to be precise). When dealing with artillery fire, the 2% discrepancy isn't that important!

• 1 degree = .017453 radians = 1/57.3 (1/60 is a good approximation)
• 1' = .0002909 = 1/3400
• 1" = .0000048481 = 1/206,000

When dealing with very small angles, the following approximations are very useful.

• sin A = A = tan A
• csc A = 1/A = cot A
• cos2 A = 1 - A /2
• sec2 A = 1 + A /2

These approximations are valid for all practical purposes for angles less than 1 degree and are accurate within 1% for angles less than 10 degrees

Some other useful approximations---
If x is small compared to 1:

• 1/(1 - x) = 1 + x
• 1/(1 + x) = 1 - x
• (1 + x)n = 1 + nx
• (1 - x)n = 1 - nx
• SQRT(1 + x) = 1 + x/2
• SQRT(1 - x) = 1 - x/2
• exp(x) = 1 + x
• ln (1 + x) = x
• exp(-x) = 1 - x
• ln (1 - x) = -x

Created 5 January 1999, Last Update 30 January 2012
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