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Home : Math & Science : Math Study Guides : Precalculus : Trigonometric Functions : 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) =  |
cosine(θ) = cos(θ) =  |
tangent(θ) = tan(θ) =  |
cosecant(θ) = csc(θ) =  |
secant(θ) = sec(θ) =  |
cotangent(θ) = cot(θ) =  |
The unit circle is the circle with equation x2 + y2 = 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. |
| tangent(s) = tan(s) = |
cosecant(s) = csc(s) =  |
secant(s) = sec(s) =  |
| cotangent(s) = cot(s) = |
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) =  |
cosine(θ) = cos(θ) =  |
tangent(θ) = tan(θ) =  |
cosecant(θ) = csc(θ) =  |
secant(θ) = sec(θ) =  |
cotangent(θ) = cot(θ) =  |
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.
Figure 2.1: The signs of the trigonometric functions in the quadrants
The quadrantal angles have the values as shown in the chart below.
Figure 2.2: The values of the trigonometric functions of quadrantal angles
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.
Figure 2.3: In each drawing, β is the reference angle for θ.
Figure 2.4: How to calculate the reference angle β for any angle
θ between 0 and 2π radians.
It will be useful to memorize the values of the trigonometric functions at a few
common angles besides the quadrantal angles: θ = 30o, 45o, 60circ. Use the values below to find the values of cosecant, secant,
and cotangent at these angles.
values
sin(30o) =  |
cos(30o) = frac |
tan(30o) =  |
sin(45o) =  |
cos(45o) = frac1 |
| tan(45o) = 1. |
sin(60o) =  |
| cos(60o) = frac12 |
| tan(60o) = |
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