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
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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.
Figure %: The signs of the trigonometric functions in the quadrants
The quadrantal angles have the values as shown in the chart below.
Figure %: 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 %: In each drawing,
β
is the reference angle for
θ
.
Figure %: 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 = 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
) =  .
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cos(30
o
) = frac
 2.
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tan(30
o
) =  .
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sin(45
o
) =  .
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cos(45
o
) = frac1 .
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sin(60
o
) = 
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|
tan(60
o
) =
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