Steven Dutch, Professor Emeritus, Natural and Applied Sciences, University of Wisconsin - Green Bay
Trigonometric functions are simply the ratios between sides of a right triangle
We can also define these less-used functions:
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
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 |
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 |
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!
When dealing with very small angles, the following approximations are very useful.
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:
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Created 5 January 1999, Last Update 12 June 2020