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Section 3.1
Radian Measure
Radian measure enables us to treat the trigonometric functions as functions with domains (inputs) of REAL NUMBERS,
rather than angles.
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.
If 𝜽 is a central angle of a circle of radius 𝒓,
and 𝜃 intercepts an arc of length s, then the
radian measure of 𝜃 is found by,
𝜽=
How many radians represents a complete rotation?
𝒔
𝒓
Converting between degrees and radians
Since 𝟏𝟖𝟎° = 𝝅 radians ,
1 degree =
radians
1 radian =
and
degrees
To convert from
1) degrees to radians, multiply by
2) radians to degrees, multiply by
Example 1: Convert each degree measure to radians. Show as a multiple of 𝜋 and give its decimal approximation.
a) 60°
b) −135°
c) 325.7°
Example 2: Convert each radian measure to degrees.
a) 2.92
NOTE: If no unit of measurement is specified for an
angle, then the angle is understood to be in radians.
NOTE: A shortcut to convert radians to degrees: substitute 180° for 𝜋.
b) 13𝜋
c) 3 Important degree measures and their radian equivalences
Reference angles are super easy to recognize for
the radian measures shown here.
,
,
,
,
,
,
Reference angle:
,
,
,
Reference angle:
𝜃
Reference angle:
sin 𝜃
cos 𝜃
tan 𝜃
cot 𝜃
sec 𝜃
csc 𝜃
𝜋
1
2
√3
2
√3
3
√3
2√3
3
2
𝜋
√2
2
1
1
√2
√2
𝜋
√3
2
√2
2
1
2
√3
√3
3
2
2√3
3
30°= 6
45°= 4
60°= 3
Example 3: Find the exact value of each expression without using a calculator.
𝑎) tan
= _________
b) sec
= ________
Quadrant:
Quadrant:
Reference Angle:
Reference Angle:
c) sin −
=________
Quadrant:
Reference Angle:
d) cos 3𝜋 = ________
e) sin −
Quadrant:
Reference Angle:
=________