Trigonometry Refresher
Steven Dutch, Professor Emeritus, Natural and Applied Sciences, University of Wisconsin - Green Bay
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!
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:
- 1 degree = .017453 radians = 1/57.3 (1/60 is a good approximation)
- 1' = .0002909 = 1/3400
- 1" = .0000048481 = 1/206,000
- sin A = A = tan A
- csc A = 1/a = cot A
- cos2 A = 1 - A /2
- sec2 A = 1 + A /2
- 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 12 June 2020