Precalculus: Trigonometric Functions
Trigonometric Functions
There are three basic ways to define the trigonometric functions.
Consider a point (x, y) on the terminal side of an angle θ in standard position. It lies a distance d away from the origin.
trigfunctions
sine(θ) = sin(θ) =  .
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cosine(θ) = cos(θ) =  .
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tangent(θ) = tan(θ) =  .
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cosecant(θ) = csc(θ) =  .
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secant(θ) = sec(θ) =  .
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cotangent(θ) = cot(θ) =  .
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The unit circle is the circle with equation x 2 + y 2 = 1 . Let s be the length of the arc with one endpoint at (1, 0) extending around the circle counterclockwise with its other endpoint at (x, y) . Note that s is both the length of an arc as well as the measure in radians of the central angle that intercepts that arc.
trigfunctions2
| sine(s) = sin(s) = y. |
| cosine(s) = cos(s) = x. |
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tangent(s) = tan(s) = ;;x≠0.
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cosecant(s) = csc(s) =  ;;y≠0.
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secant(s) = sec(s) =  ;;x≠0.
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cotangent(s) = cot(s) = ;;y≠0.
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Consider a right triangle with one acute angle θ in standard position. Let the side opposite that angle be called the opposite side. Let the other leg be called the adjacent side.
trigfunctions3
sine(θ) = sin(θ) = 
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cosine(θ) = cos(θ) = 
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tangent(θ) = tan(θ) = 
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cosecant(θ) = csc(θ) = 
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secant(θ) = sec(θ) = 
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cotangent(θ) = cot(θ) = 
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The trigonometric functions have different signs according to the quadrant in which the angle's terminal side lies. Here is a chart to show these signs.
The quadrantal angles have the values as shown in the chart below.
A reference angle is the positive acute angle created by the terminal side of an angle in standard position and the x -axis. A reference angle, therefore, is a first quadrant angle. Let β be the reference angle of θ . The value of a trigonometric function at θ is equal to the value of that function at β -- unless there is a variation in sign. The sign difference depends on which quadrant θ is in. An understanding of reference angles simplifies the evaluation of trigonometric functions of large angles.
It will be useful to memorize the values of the trigonometric functions at a few common angles besides the quadrantal angles: θ = 30 o , 45 o , 60c irc . Use the values below to find the values of cosecant, secant, and cotangent at these angles.
values
sin(30
o
) =  .
|
cos(30
o
) = frac
 2.
|
tan(30
o
) =  .
|
sin(45
o
) =  .
|
cos(45
o
) = frac1 .
|
| tan(45 o ) = 1. |
sin(60
o
) = 
|
| cos(60 o ) = frac12 |
|
tan(60
o
) =
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