MAC 1114

1
MAC 1114

• Module 3
• Radian Measure and
• Circular Functions

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Learning Objectives

• Upon completing this module, you should be able
  to
• Convert between degrees and radians.
• Find function values for angles in radians.
• Find arc length on a circle.
• Find area of a sector of a circle.
• Solve applications.
• Define circular functions.
• Find exact circular function values.
• Approximate circular function values.

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Rev.S08
3
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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Rev.S08
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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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Rev.S08
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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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Rev.S08
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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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Example of Converting from Radians to Degrees

• Convert each radian measure to degrees.
• a)
• b) 3.25

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Rev.S08
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Lets Look at Some Equivalent Angles in Degrees
and Radians
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Rev.S08
9
Lets Look at Some Equivalent Angles in Degrees
and Radians (cont.)
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Examples

• Find each function value.
• a)
• Convert radians to degrees.

• b)

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Rev.S08
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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 ?
  radians is given by the product of the radius and
  the radian measure of the angle, or s r?, ? in
  radians.

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Rev.S08
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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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Example of Finding Arc Length of a Circle (cont.)

• a) r 18.2 cm and ?

• 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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Lets 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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Rev.S08
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Lets 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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Rev.S08
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Lets Practice Another Application of Radian
Measure Problem (cont.)

• An arc with this length on the larger gear
  corresponds to an angle measure ?, in radians
  where
• Convert back to degrees.

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Rev.S08
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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 ? is given by

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Example

• Find the area of a sector with radius 12.7 cm and
  angle ? 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 r?,
• s? 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 ? measured in
radians, because r 1 in the unit circle.
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Rev.S08
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Lets 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 (-8, 8)
• 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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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 ApproximatingCircular 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
• Convert between degrees and radians.
• Find function values for angles in radians.
• Find arc length on a circle.
• Find area of a sector of a circle.
• Solve applications.
• Define circular functions.
• Find exact circular function values.
• Approximate circular function values.

http//faculty.valenciacc.edu/ashaw/ Click link
to download other modules.
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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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Rev.S08