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MAT 170 Pre-Calculus
Radian Angles
Notes on Radian Measure
Terri L. Miller
Spring 2009
revised April 17, 2009
1. Radian Measure
Recall that a unit circle is the circle centered at the origin with a radius of one. Since the
radius is one, the distance around the circle, the circumference, is 2π.
The terminal side of an angle cuts the unit circle at a point, P (x, y). The measure of
the distance in moving from (1, 0) to this point, P (x, y) along the unit circle in a counterclockwise direction is the radian measure of the angle, a positive number. Note: any real
number can be a radian measure! So a negative number would indicate to move the distance
from (1, 0) to the point in a clockwise direction.
For the unit circles pictured below, we have indicated by a dot, the point P (x, y) where the
terminal side of an angle would cut the circle. The angle is shaded and the measurement
indicated.
Figure 1. The angles of radian measures 1 and 2
Here C marks the terminal point on the unit circle for the angle α = 1 and D marks the
terminal point on the unit circle for the angle β = 2.
Here D marks the terminal point on the unit circle for the angle θ = 4 and E marks the
π
terminal point on the unit circle for the angle δ = − .
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Let us consider some standard angles on the unit circle. First, recall that the distance around
the unit circle is 2π. Hence θ = 2π is one full circuit of the unit circle so that both the initial
and terminal points are (1, 0). Going half way around the circle would be a distance of
2π
= π, so the terminal point would be (−1, 0).
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Here C marks the terminal point on the unit circle for the angle θ = 2π and B0 marks the
terminal point on the unit circle for the angle α = π.
Figure 2. The angles of radian measures 4 and −
π
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Figure 3. The angle 2π and the angle π
π π π π
The standard angles to mark off are the multiples of 0, , , , , and π. These are chosen
6 4 3 2
since we do know the exact coordinates of their terminal points. However, we will save the
coordinates for a bit later.
Figure 4. Some standard angles.
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c 2009 ASU School of Mathematical & Statistical Sciences and Terri L. Miller
π
π
π
2π
7π
7π
,β = ,γ = ,δ =
, =
, and δ =
.
6
4
3
3
6
4
Here are some more standard but negative angles.
The angles pictured here are α =
Figure 5. Some negative angles.
π
5π
−5π
5π
π
, and = − .
The angles pictured here are α = − , β = − , γ = − , δ =
4
2
6
4
3
We can also mark off any real number.
Figure 6. Arbitrary numbers
The angles pictured here are α = 0.5, β = 2.7, γ = −
3
15
, and δ = −3.
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c 2009 ASU School of Mathematical & Statistical Sciences and Terri L. Miller
2. The Coordinates
We know the coordinates for all of the points along the axes. Further, we know that every
point on the unit circle satisfies the equation x2 + y 2 = 1. The angle π4 cuts the unit circle in
the middle of the first quadrant. This corresponds to the line y = x, The coordinates of this
point can easily be obtained by solving the equations
y = x and x2 + y 2 = 1 for x. When
√ √ !
2 2
this is done, we find the coordinates are
,
.
2 2
Figure 7. Coordinates on the Unit Circle
In the left hand figure, the points along the axes are given. In the right hand figure, the
points of intersection with the lines!y = x and y = −x have
been marked. The!coordinates
√ √
√
√ !
√ √
2 2
2
2
2 2
for these point are F =
,
,G = −
,−
,H = −
,
, and J =
2 2
2
2
2 2
√
√ !
2
2
,−
.
2
2
The use of equilateral triangles and some basic geometry yields the coordinates for the points
corresponding to the terminal sides of the angles π6 and π3 and hence their multiples in other
quadrants. These are all labelled later on a single unit circle.
3. The Functions
The cosine is the x coordinate of the point on the unit circle corresponding to the terminal
side of an angle. The function is written cos θ. The sine is the y coordinate of the point on
the unit circle corresponding to the terminal side of an angle. The sine function is written
sin θ.
The other four trigonometric functions are defined in terms of the sine and cosine, the
tangent is the ration of the sine over the cosine (denoted tan θ), the secant is the reciprocal
of the cosine (denoted sec θ), the cosecant is the reciprocal of the sine (denoted csc θ), and
the cotangent is the ratio of the cosine over the sine (denoted cot θ).
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c 2009 ASU School of Mathematical & Statistical Sciences and Terri L. Miller
Figure 8. The Unit Circle
4. Links
For further reading, I recommend checking out the following url:
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c 2009 ASU School of Mathematical & Statistical Sciences and Terri L. Miller
http://www.themathpage.com/atrig/radian-measure.htm
For further unit circle help, try the following url:
http://members.shaw.ca/ron.blond/TLE/WRAPPING.FUNCTION.APPLET/index.html
or
http://www.mathlearning.net/learningtools/Flash/unitCircle/unitCircle.html
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c 2009 ASU School of Mathematical & Statistical Sciences and Terri L. Miller