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This chapter was remixed from Precalculus: An Investigation of Functions , (c) 2013 David Lippman and Melonie Rasmussen. It is licensed under the Creative Commons Attribution license.
Section 9: Trigonometric Functions
When working with triangles and circles, we commonly need to work with angles.
Measure of an Angle The measure of an angle is a measurement between two intersecting lines, line segments or rays, starting at the initial side and ending at the terminal side. It is a rotational measure not a linear measure.
There are two common ways to measure angles: using degrees and radians. When measuring angles on a circle, unless otherwise directed, we measure angles in standard position : starting at the positive horizontal axis and with counter-clockwise rotation.
Degrees A degree is a measurement of angle. One full rotation around the circle is equal to 360 degrees, so one degree is 1/360 of a circle.
An angle measured in degrees should always include the unit “degrees” after the number, or include the degree symbol °. For example, 90 degrees = 90 °.
Radians are a different measure of angle based on arclength. On a unit circle, a circle with radius 1, the arclength subtended by an angle θ corresponds to the radian measure. In general,
Radians The radian measure of an angle is the ratio of the length of the circular arc subtended by the angle to the radius of the circle.
In other words, if s is the length of an arc of a circle, and r is the radius of the circle, then
radian measure
s r
If the circle has radius 1, then the radian measure corresponds to the length of the arc.
Because radian measure is the ratio of two lengths, it is a unitless measure. It is not necessary to write the label “radians” after a radian measure, and if you see an angle that is not labeled with “degrees” or the degree symbol, you should assume that it is a radian measure.
Considering the most basic case, the unit circle (a circle with radius 1), we know that 1 rotation equals 360 degrees, 360 °. We can also track one rotation around a circle by finding the
circumference, C = 2 π r , and for the unit circle C = 2 π. These two different ways to rotate
around a circle give us a way to convert from degrees to radians.
initial side
terminal side
angle
1 rotation = 360 ° = 2 π radians
½ rotation = 180 ° = π radians
¼ rotation = 90 ° = π / 2 radians
Converting Between Radians and Degrees
1 degree = 180
radians
or: to convert from degrees to radians, multiply by
radians 180
1 radian =
degrees
or: to convert from radians to degrees, multiply by
πradians
Example 1
Convert 6
radians to degrees.
Since we are given a problem in radians and we want degrees, we multiply by
Remember radians are a unitless measure, so we don’t need to write “radians.”
radians = 30
degrees.
Example 2 Convert 15 degrees to radians.
In this example we start with degrees and want radians so we use the other conversion 180 °
π so
that the degree units cancel and we are left with the unitless measure of radians.
15 degrees = 180 12
In calculus, unless otherwise stated, we always measure angles in radians.
Example 4 Find cos( 90 °) and sin( 90 °)
On any circle, the terminal side of a 90 degree angle points straight up, so the coordinates of the corresponding point on the circle would be (0, r ). Using our definitions of cosine and sine,
0
cos( 90 °) = = = r r
x
sin( 90 °) = = = 1 r
r r
y
Usually we have to use a calculator to evaluate the values of sine and cosine, but there are a few angles that have nice exact values.
Angle 0
6
, or 30° 4
, or 45° 3
, or 60° 2
, or 90°
Cosine (^1 )
2
Sine 0 1
2
A full circle with all the “special” angles and coordinates can be found at the end of the section.
Notice that the circle definitions above can also be stated as:
Coordinates of the Point on a Circle at a Given Angle
On a circle of radius r at an angle of θ , we can find the coordinates of the point ( x , y ) at
that angle using
x = r cos( θ )
y = r sin( θ )
On a unit circle, a circle with radius 1, x =cos( θ) and y =sin( θ).
Example 5
Find the coordinates of the point on a circle of radius 12 at an angle of 6
Note that this angle is in the third quadrant, where both x and y are negative. Keeping this in mind can help you check your signs of the sine and cosine function.
r 90°
(0, r )
12 cos =−
π x
12 sin =−
y
The coordinates of the point are ( − 6 3 ,− 6 ).
Tangent, Secant, Cosecant, and Cotangent
In addition to sine and cosine, there are four other common trigonometric functions that can be defined in terms of sine and cosine.
Tangent, Secant, Cosecant, and Cotangent Functions For the point ( x , y ) on a circle of radius r at an angle of θ , we can define four additional important functions as the ratios of the sides of the corresponding triangle. These can also be expressed in terms of sine and cosine.
The tangent function: x
y
tan( θ ) =
cos( )
sin( ) tan( )
The secant function: x
r sec( θ ) = cos( )
sec( )
The cosecant function: y
r csc( θ ) = sin( )
csc( ) θ
θ =
The cotangent function: y
x
cot( θ ) =
1 cos( ) cot( ) tan( ) sin( )
Example 6
Evaluate tan( 45 °) and (^)
sec
Since we know the sine and cosine values for these angles, it makes sense to relate the tangent and secant values back to the sine and cosine values.
cos( 45 )
sin( 45 ) tan( 45 ) = = °
( x , y )
r
θ
y
x
Notice how the sine values are positive between 0 and π, which correspond to the values of sine in quadrants 1 and 2 on the unit circle, and the sine values are negative between π and 2π, corresponding to quadrants 3 and 4.
Like the sine function we can track the value of the cosine function through the 4 quadrants of the unit circle as we place it on a graph.
Both of these functions are defined for all real numbers, since we can evaluate the sine and cosine of any angle. By thinking of sine and cosine as coordinates of points on a unit circle, it becomes clear that the range of both functions must be the interval (^) [− 1 , 1 ]. The period of both
functions, the time it takes to complete one full cycle, is 2π.
Domain and Range of Sine and Cosine The domain of sine and cosine is all real numbers, (^) ( −∞ ∞, ). The range of sine and cosine is the interval [-1, 1]. The periods of the sine and cosine functions are both 2π.
Using transformations, we can sketch other sinusoidal functions.
f(θ) = sin( θ )
θ
θ
g(θ) = cos( θ )
Transformations of Sine and Cosine
Given an equation in the form f ( t )= A sin( B ( t − h )) + k or f ( t )= A cos( B ( t − h )) + k
A is the vertical stretch, and is the amplitude of the function.
B is the horizontal stretch/compression, and is related to the period, P , by B
P
k is the vertical shift and determines the midline of the function. h is the horizontal shift of the function
Example 8 Find a formula for the sinusoidal function graphed here.
The graph oscillates from a low of -1 to a high of 3, putting the midline at y = 1, halfway between.
The amplitude will be 2, the distance from the midline to the highest value (or lowest value) of the graph.
The period of the graph is 8. We can measure this from the first peak at x = -2 to the second at x = 6. Since the period is 8, the stretch/compression factor we will use will be
P
B
At x = 0, the graph is at the midline value, which tells us the graph can most easily be represented as a sine function. Since the graph then decreases, this must be a vertical reflection of the sine function. Putting this all together,
1 4
( ) 2 sin +
f t = − t
Example 9
Sketch a graph of (^)
() 3 sin
π π f t t.
To reveal the horizontal shift, we first need to factor inside the function:
f ( t ) 3 sin t
This graph will have the shape of a sine function, starting at the midline and increasing, with
an amplitude of 3. The period of the graph will be 8
B
P. Finally, the
graph will be shifted to the right by 1.
