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General Angles and Radian Measure
Goal: To introduce the concept of angles of any measure. The measure of an angle
is no longer restricted to those angles within a triangle we now look at angles whose
measure can be any real number. We will look at two units of measure for angles,
the radian measure of angles as well as degrees; our study will focus on
understanding the different units of measure and learning how to convert from
radian to degree and degree to radians.
First let us recall from geometry, an angle, is formed by two rays that have a
common endpoint called the vertex. You can generate any angle by fixing one ray
called the initial side, and rotating the other ray called the terminal side, around the
vertex.
***On Board Angles of any measure/Algebra in Motion Demo
In a coordinate plane an angle whose vertex is at the origin and whose initial side is
the positive x-axis is said to be in standard position.
The measure of an angle is determined by the amount and direction of rotation from
the initial side to the terminal side.
The angle measure is positive if the direction is counterclockwise, and negative if the
rotation is clockwise.
One complete rotation around the vertex is 360 degrees.
One half revolution is 180 degrees
One-quarter revolution is 90 degrees.
Three – quarter of a revolution is 270 degrees
Define the quadrants:
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Quadrant I (0-90 degrees)
Quadrant II (90-180 degrees)
Quadrant III (180- 270 degrees)
Quadrant IV (270 –360 degrees)
You need to be able to draws an angle in standard position and tell which quadrant
the terminal side lies in.
If the terminal side of an angle in standard position lies on either the positive or
negative x or y-axis the angle is called a quadrantal angle.
Show 0, 90, 180, 270 and 360 degree angles using Algebra in motion.
Co Terminal Angles:
Show 510 degrees use this angle to illustrate that 510 and 150 degrees angles are coterminal.
Two angles in standard position are co-terminal if they share terminal sides (are the
same).
Adding or subtracting multiples of 360 degrees can find any angle co-terminal with
a given angle.
Finding co-terminal angles:
Example: Find one positive co-terminal angle with 45 degrees.
Solution: There are as many answers as there are multiples of 360 degrees. One
example is 45 + 360 = 205 degree angle
Example 2: Find one positive and one negative angle
That are co- terminal with - 60 degrees angle
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Positive co-terminal angle: -60 + 360 = 300 degrees
Negative co-terminal angle: -60 –360 = -420 degrees
**Note there are infinitely many co-terminal angles. However, often you will be
asked to find the smallest positive or negative co terminal angle.
Find the smallest, positive, co-terminal angle with 495 degrees:
495 – 360 = 135 degrees
Find a negative co-terminal angle with the given angle: 495 degrees
495- 360 = 135 NO you must subtract two multiples of 360: 495- 2(360) =-225
Radian measure: So far all the angles we have worked with we have measured in
degrees. There is another common measurement use to measure angles that is
radians.
Radian measurement is used frequently in calculus:
A radian is the measure of an angle in standard position whose terminal side
intercepts an arc length of length r (arc length = radius)
Means in English: If a circle of radius 1 is drawn with the vertex of an angle at its
center, then the measure of this angle in radians is the length of the arc that
subtends the angle.
Let’s look at a circle: the circumference of a circle is 2 r , there are 2  radians in a
full circle.
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Degree measurer and radian measure are therefore related by the equation:
360  2 radians or 180   radians
An important point going forward: When no unit of measurement is given or
specified, radian measure is implied. For instance,   2 means that   2 radians.
In trigonometry it will be helpful for you to memorize the equivalent degree and
radian measures of special angles in the first quadrant and for 90 degrees =

. All
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the other special angles are just multiples of these angles.
Let’s investigate the relationship between radians and degrees
Use the following rules to convert degrees to radians and radians to degrees
Conversions between Degrees and Radians


Degree to radians, multiply by

Radians to degrees, multiply by
Examples:
1) Convert 110 degrees to radians
2) Convert

radians to degrees
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180
180

Solutions:
110 
110 r 11 r




1 180
180
18
 180 180


 20
9

9
4