1. MAC 1114
Module 3
Radian Measure and
Circular Functions
Rev.S08
Learning Objectives
Upon completing this module, you should be able to:
1. Convert between degrees and radians.
2. Find function values for angles in radians.
3. Find arc length on a circle.
4. Find area of a sector of a circle.
5. Solve applications.
6. Define circular functions.
7. Find exact circular function values.
8. Approximate circular function values.
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Radian Measure and Circular Functions
There are three major topics in this module:
- Radian Measure
- Applications of Radian Measure
- The Unit Circle and Circular Functions
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2. Introduction to Radian Measure
 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.
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How to Convert Between Degrees and
Radians?
 1. Multiply a degree measure by radian and
simplify to convert to radians.
 2. Multiply a radian measure by and simplify
to convert to degrees.
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Example of Converting from
Degrees to Radians
 Convert each degree measure to radians.
 a) 60°
 b) 221.7°
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3. Example of Converting from
Radians to Degrees
 Convert each radian measure to degrees.
 a)
 b) 3.25
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Let’s Look at Some Equivalent Angles in
Degrees and Radians
Degrees Radians Degrees Radians
Exact Approximate Exact Approximate
0° 0 0 90° 1.57
30° .52 180° π 3.14
45° .79 270° 4.71
60° 1.05 360° 2π 6.28
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Let’s Look at Some Equivalent Angles in
Degrees and Radians (cont.)
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4. Examples
 Find each function value.  b)
 a)
 Convert radians to
degrees.
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How to Find Arc Length of a Circle?
 The length s of the arc
intercepted on a circle of
radius r by a central angle
of measure θ radians is
given by the product of
the radius and the radian
measure of the angle, or
s = rθ, θ in radians.
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Example of Finding Arc Length of a Circle
 A circle has radius 18.2
cm. Find the length of the
arc intercepted by a
central angle having each
of the following measures.
 a)
 b) 144°
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5. Example of Finding Arc Length of a Circle
(cont.)
 a) r = 18.2 cm and θ =  b) convert 144° to radians

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Example of Application
 A rope is being wound around a
drum with radius .8725 ft. How
much rope will be wound around
the drum it the drum is rotated
through an angle of 39.72°?
 Convert 39.72 to radian
measure.
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Let’s Practice Another Application of
Radian Measure Problem
 Two gears are adjusted
so that the smaller gear
drives the larger one, as
shown. If the smaller gear
rotates through 225°,
through how many
degrees will the larger
gear rotate?
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6. Let’s Practice Another Application of
Radian Measure Problem (cont.)
 Find the radian measure of the angle and then
find the arc length on the smaller gear that
determines the motion of the larger gear.
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Let’s Practice Another Application of
Radian Measure Problem (cont.)
 An arc with this length on the larger gear
corresponds to an angle measure θ, in radians
where
 Convert back to degrees.
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How to Find Area of a Sector of a Circle?
 A sector of a circle is a portion of the interior of a
circle intercepted by a central angle. “A piece of
pie.”
 The area of a sector of a circle of radius r and
central angle θ is given by
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7. Example
 Find the area of a sector with radius 12.7 cm and
angle θ = 74°.
 Convert 74° to radians.
 Use the formula to find the area of the sector of a
circle.
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What is a Unit Circle?
 A unit circle has its center at the origin and a
radius of 1 unit.
Note: r = 1
 s = rθ,
 s=θ in radians.
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Circular Functions
Note that s is the arc length
measured in linear units such as
inches or centimeters, is
numerically equal to the angle θ
measured in radians, because r =
1 in the unit circle.
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8. Let’s Look at the Unit Circle Again
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What are the Domains of the
Circular Functions?
 Assume that n is any integer and s is a real
number.
 Sine and Cosine Functions: (−∞, ∞)
 Tangent and Secant Functions:
 Cotangent and Cosecant Functions:
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How to Evaluate a Circular Function?
 Circular function values of real numbers are
obtained in the same manner as trigonometric
function values of angles measured in radians.
This applies both to methods of finding exact
values (such as reference angle analysis) and to
calculator approximations. Calculators must be in
radian mode when finding circular function
values.
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9. Example of Finding Exact Circular
Function Values
 Find the exact values of
 Evaluating a circular function at the real number
is equivalent to evaluating it at radians. An
angle of intersects the unit circle at the point
.
 Since sin s = y, cos s = x, and

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Example of Approximating
Circular Function Values
 Find a calculator approximation to four decimal
places for each circular function. (Make sure the
calculator is in radian mode.)
 a) cos 2.01 ≈ −.4252 b) cos .6207 ≈ .8135
 For the cotangent, secant, and cosecant functions
values, we must use the appropriate reciprocal
functions.
 c) cot 1.2071
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What have we learned?
We have learned to:
1. Convert between degrees and radians.
2. Find function values for angles in radians.
3. Find arc length on a circle.
4. Find area of a sector of a circle.
5. Solve applications.
6. Define circular functions.
7. Find exact circular function values.
8. Approximate circular function values.
http://faculty.valenciacc.edu/ashaw/
Rev.S08 Click link to download other modules. 27
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10. Credit
Some of these slides have been adapted/modified in part/whole from the
slides of the following textbook:
• Margaret L. Lial, John Hornsby, David I. Schneider, Trigonometry, 8th
Edition
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