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

                                X
                               /|
                              / |
                             /  |
                 Hypotenuse /   | Opposite Side
                           /    |
                          /     |
                         /      |
                 Angle A/_______|
                        Adjacent Side

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

We can also define these less-used functions:

Cosecant = Hypotenuse / Opposite = 1 / Sine
Secant = Hypotenuse / Adjacent = 1 / Cosine
Cotangent = Adjacent / Opposite = Cosine / Sine = 1 / Tangent

The Pythagorean Relations

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


                                X
                               /|
                              / |
                             /  |
                          1 /   | Sin A
                           /    |
                          /     |
                         /      |
                 Angle A/_______|
                           Cos A

From the Pythagorean Theorem, it is obvious that

                       2           2
                    Sin A   +   Cos A   =   1

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

         2          2              2               2
1  +  Cot A  =  Csc  A          Tan A  +  1  =  Sec  A

These are the Pythagorean Relations

Relations Between Functions

We can just as easily define our angles this way


                                X
                               /| Angle B
                              / |
                             /  |
                          1 /   | Cos B
                           /    |
                          /     |
                         /      |
                        /_______|
                          Sin B

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:

                                |
                     (x,y) *    |
                            \   |            Sin, Cos, Tan +
         Sin +               \  |
         Cos, Tan -           \ |
                               \|A
                      ----------|----------  
                                |
                                |
                                |
         Tan +, Sin, Cos -      |            Cos +, Sin, Tan -
                                |
x = Cos A,   y = Sin A          |
                                |

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

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

sin A = A = tan A
csc A = 1/A = cot A
2
cos A = 1 - A /2
2
sec 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

60. 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

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Created 5 January 1999, Last Update 5 January 1999
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