rev.s08 mac 1114 module 3 radian measure and circular functions

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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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http://faculty.valenciacc.edu/ashaw/

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Radian Measure and Circular Functions


- Radian Measure- Radian Measure

- Applications of Radian Measure- Applications of Radian Measure

- The Unit Circle and Circular Functions- The Unit Circle and Circular Functions

There are three major topics in this module:


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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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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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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


6.282π3601.0560

4.71270.7945

3.14π180.5230

1.5790000

ApproximateExactApproximateExact

RadiansDegreesRadiansDegrees


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Let’s 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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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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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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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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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

θmeasured in radians, because r = 1 in the unit circle.

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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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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.


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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


rev.s08 mac 1114 module 3 radian measure and circular functions

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