Radians and Degrees

TrigonometryMATH 103

S. Rook

Overview

• Section 3.2 in the textbook:– The radian– Converting between degrees and radians– Radians and Trigonometric functions– Approximating with a calculator

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The Radian

Motivation for Introducing Radians

• Thus far we have worked exclusively with angles measured in degrees

• In some calculations, we require theta 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– Informally: How many radii r comprise the arc length s

• For θ = 1 radian, s = r

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r

s

Radians (Example)

Ex 1: Find the radian measure of θ, if θ is a central angle in a circle of radius r, and θ cuts off an arc of length s:

r = 10 inches, s = 5 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 circle8

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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

• 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

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180

rad

Converting from Degrees to Radians (Example)

Ex 2: For each angle θ, i) draw θ in standard position, ii) convert θ to radian measure using exact values, iii) name the reference angle in both degrees and radians:

a) -120°b) 390°

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Converting from Radians to Degrees

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

so that radians will divide out leaving degrees

• The concept of reference angles still applies when θ is in radians: – Instead of adding/subtracting 180° or 360°,

add/subtract π or 2π respectively

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rad

180

Converting from Radians to Degrees (Example)

Ex 3: For each angle θ, i) draw θ in standard position, ii) convert θ to degree measure, iii) name the reference angle in both radians and degrees:

a)

b)

13

6

11

4

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Radians and Trigonometric Functions

Radians and Trigonometric Functions

• On the next slide is a table of values you should have already memorized when θ was in degrees

• Only difference is the equivalent radian values– Each radian value can be obtained via the

conversion procedure previously discussed

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Radians and Trigonometric Functions (Continued)

Degrees Radians cos θ sin θ

0° 0 1 0

30°

45°

60°

90° 0 1

180° -1 0

270° 0 -1

360° 1 0

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6

4

3

2

2

3

2

2

3

2

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2

2

1

2

2

2

1

2

1

2

1

Radians and Trigonometric Functions (Example)

Ex 4: Give the exact value:

a)

b)

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3

5cos

6

7csc

Radians and Trigonometric Functions (Example)

Ex 5: Find the value of y that corresponds to each value of x and then write each result as an ordered pair (x, y):

y = cos x for

18

,

4

3,

2,

4,0x

Approximating with a Calculator

Approximating with a Calculator

• When approximating the value of a trigonometric function in radians:– Ensure that the calculator is set to radian mode

• ESSENTIAL to know when to use degree mode and when to use radian mode:– Angle measurements in degrees are post-fixed with

the degree symbol (°)– Angle measurements in radians are sometimes given

the post-fix unit rad but more commonly are given with no units at all

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Approximating with a Calculator (Example)

Ex 6: Use a calculator to approximate:

a) sin 1

b) cos 3π

c)

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7

2cot

Summary• After studying these slides, you should be able to:– Calculate the radian measure of a circle with radius r

cutting off an arc length of s– Convert between degrees & radians and vice versa– Apply previous concepts such as reference angles to

angles measured in radians– Use a calculator to approximate the value of a

trigonometric function in both degrees and radians• Additional Practice– See the list of suggested problems for 3.2

• Next lesson– Definition III: Circular Functions (Section 3.3)

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