Radian & Degree Measure

MATH 109 - PrecalculusS. Rook

Overview

• Section 4.1 in the textbook:– Angles– Degree measure– Radian measure– Converting between degrees & radians

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Angles

Angles

• Angle: describes the “space” between two rays that are joined at a common endpoint– Recall from Geometry that a ray has one

terminating side and one non-terminating side

• Can also think about an angle as a rotation about the common endpoint– Start at OA (Initial side)– End at OB (Terminal side)

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Angles (Continued)

• If the initial side is rotated counter-clockwise

θ is a positive angle

• If the initial side is rotated clockwise

θ is a negative angle

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Angles in Standard Position

• An angle θ is in standard position if its:– Initial side extends along the positive x-axis

in reference to the Cartesian Plane– Vertex is (0, 0)

• The “element of” symbol can be used to denote an angle in standard position– e.g. means θ is in standard position

with its terminal side in Quadrant III

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QIII

Degree Measure

Angle Measure

• Angle Measure: expresses the size of an angle– i.e. the space in between the initial and terminal

sides in the direction of rotation

• Two common types of angle measures:– Degrees– Radians

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Degree Measure

• 1 degree corresponds to (1⁄360) of a complete revolution starting from the initial side of an angle to its terminal side– i.e. Can be viewed in terms of a circle

• Common degree measurements to be familiar with: 360° makes one complete revolution• The initial and terminal sides of the angle are the same

180° makes one half of a complete revolution90° makes one quarter of a complete revolution

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Degree Measure (Continued)

• Angles that measure:– Between 0° and 90° are known as acute angles– Exactly 90° are known as right angles• Denoted by a small square between the initial and

terminal sides– Between 90° and 180° are known as obtuse angles

• Complementary angles: two angles whose measures sum to 90°

• Supplementary angles: two angles whose measures sum to 180°

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Degrees & Minutes• Degrees can be broken down even further using minutes

1° = 60’• To convert from decimal degrees to degrees and minutes:

– Use the decimal portion of the angle– Multiply by the appropriate conversion ratio

• Align the units in the ratio so the degrees will divide out, leaving the minutes

• To convert from degrees and minutes to decimal degrees:– Use the minutes from the angle measurement– Multiply by the appropriate conversion ratio

• Align the units in the ratio so the minutes will divide out, leaving the degrees

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Sketching Angles in Standard Position (Example)

Ex 1: Sketch each angle in standard position:

a) 293°

b) -115°

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Complementary & Supplementary Angles (Example)

Ex 2: Find: i) the complement ii) the supplement

θ = 65°

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Converting from Degrees to Minutes & Vice Versa (Example)

Ex 3: Convert a) to degrees and minutes and convert b) to decimal degrees – approximate if necessary:

a) θ = 232.55°

b) θ = 17° 22’

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Radian Measure

Motivation for Introducing Radians

• In some calculations, we require the measure of an angle (θ) to be a real number – we need a unit other than degrees– This unit is known as the radian

• Many calculations tend to become easier to perform when θ is in radians– Further, some calculations can be performed or even

simplified ONLY if θ is in radians– However, degrees are still in use in many applications so

a knowledge of both degrees and radians is ESSENTIAL

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Radians

• Radian Measure: A circle with central angle θ and radius r which cuts off an arc of length s has a central angle measure of where θ is in radians

– i.e. How many radii r comprise the arc length s

• For θ = 1 radian, s = r

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r

s

Radian Measure (Example)

Ex 4: Find the radian measure of the central angle of a circle of radius r that subtends an arc length of s

A radius of 27 inches and an arc length of 6 inches

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Converting Between Degrees and Radians

Relationship Between Degrees and Radians

• Given a circle with radius r, what arc length s is required to make one complete revolution?– Recall that the circumference measures the distance or

length around a circle– What is the circumference of a circle with radius r?

C = 2πr

• Thus, s = 2πr is the arc length of one revolution and is the number of radians in one

revolution• Therefore, θ = 360° = 2π consists of a complete revolution

around a circle20

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r

r

r

s

Relationship Between Degrees and Radians (Continued)

• Equivalently: 180° = π radians– You MUST memorize this conversion!!!

• Technically, when measured in radians, θ is unitless, but we sometimes append “radians” to it to differentiate radians from degrees– Like radians, real numbers are unitless as well

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Converting from Degrees to Radians & Vice Versa

• To convert from degrees to radians:– Multiply by the conversion ratio

so that degrees will divide out leaving radians

– If an exact answer is desired, leave π in the final answer– If an approximate answer is desired, use a calculator to

estimate π• To convert from radians to degrees:– Multiply by the conversion ratio

so that radians will divide out leaving degrees

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180

rad

rad

180

Common Angles

• Need to become familiar with the degree and radian conversion between the following commonly used angle measurements:

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Deg Rad

0° 0

30° π⁄6

45° π⁄4

60° π⁄3

90° π⁄2

180° π

270° 3π⁄2

360° 2π

Converting from Degrees to Radians & Vice Versa (Example)

Ex 5: Convert a) & b) to degrees and convert c) & d) to radians – leave π in the answer when necessary:

a) b)

c) d)

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2.4

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Coterminal Angles

• Two angles are coterminal if:– BOTH are standard angles– Share the SAME terminal side

• How can we obtain an angle coterminal to an angle θ?– The second angle must terminate where θ

terminates– Recall that one complete revolution around a

circle is 360° in degrees or 2π in radians

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Coterminal Angles (Example)

Ex 6: Do the following:

a) Given θ = -190° find in degrees: i) two coterminal angles and ii) all angles coterminal to θ

b) Given θ = π⁄8 find in radians: i) two coterminal angles and ii) all angles coterminal to θ

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Summary

• After studying these slides, you should be able to:– Draw an angle in standard position– Find both the complement and supplement of an angle– Convert between degrees & minutes and decimal degrees and vice

versa– Calculate the radian measure of a circle with radius r and

subtended by an arc length s– Convert between radians & degrees and vice versa

• Additional Practice– See the list of suggested problems for 4.1

• Next lesson– Trigonometric Functions: The Unit Circle (Section 4.2)

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