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Trigonometry Lecture Notes
Section 3.1
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Section 3.1: Radian Measure
Big Idea: The radian is an extremely convenient angle measure because of its natural connection
to circles.
Big Skill: You should be able to convert between degrees and radians.
Radian Measure
Radian measure of angles is preferred because:
 the radian measure of a central angle in a circle is the proportionality constant between
the arc length intercepted by that central angle and the radius of the circle (s = r).
 This implies that there are 2 radians in a full circle, because the arc length around a full
circle is the circumference: s = C = 2r.

radian measure allows us to treat the domain of trigonometric functions as real numbers,
rather than as angles.
Definition of the Radian
An angle with its vertex at the center of a circle that intercepts an arc on the circle equal in length
to the radius of the circle has a measure of 1 radian.
Trigonometry Lecture Notes
Section 3.1
Page 2 of 4
Converting Between Degrees and Radians
 There are 2 radians in a full circle.
 There are 360 in a full circle.
 This implies the following conversion factors:
360  2
180  
180


1

To convert from degrees to radians, multiply by

To convert from radians to degrees, multiply by
180

180
180
1
.
.

 If no unit of angle measure is stated, then the angle is understood to be measure in
radians.
 Example: in the picture on the right, the measure of the angle is understood to be radians.
Trigonometry Lecture Notes
Section 3.1
Page 3 of 4
Practice:
1. State the radian measure of all multiples of 30 and 45 going around a circle using the
notion of these angles as a fraction of a full circle.
Trigonometry Lecture Notes
2. Convert to radians: 108
3. Convert to radians: -228
4. Convert to radians: 457.89
5. Convert to degrees:
11
12
6. Convert to degrees: 
7
6
7. Convert to degrees: 15.292
 5 
8. Find the value of cos 

 6 
 
9. Find the value of sin  
4
 7 
10. Find the value of tan 

 6 
11. Find the value of sec  
 7 
12. Find the value of csc 

 4 
 2 
13. Find the value of cot  

 3 
Section 3.1
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