There are three basic ways to
define the
trigonometric functions.
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.
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 = 30o, 45o, 60circ. Use the values below to find the values of cosecant, secant,
and cotangent at these angles.