Angles & Angle Measures

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1Notation, Definitions& Measurement of Angles,

Coterminal, Right, Complementary, Supplementary Angles & Intro to Radians

Practice Problems

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Notation

Variables for angles Frequently Greek letters α (alpha) β (beta) γ (gamma) θ (theta)

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Definitions

Initial side Point of origin for measuring a given angle Typically 0˚ (360˚)

Terminal Side Ending point for measuring a given angle Can be any size

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Measurement

Clockwise (CW) Negative Angle

Counter-Clockwise (CCW) Positive Angle

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Measurement (Cont.)

Degrees May be in decimal form (72.64˚) May be in Degrees/Minutes/Seconds (25˚

43’ 37”) Minutes ( ’ ) 60’ = 1˚ Seconds ( ” ) 60” = 1’

90˚ = 89˚ 59’ 60”

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Measurement (Cont.)

Radians Similar to degrees Always measured in terms of pi (π)

360˚/0˚ = 2π 90˚ = π/2 180˚ = π 270˚ = 3 π/2

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Coterminal Angles

Have the same initial and terminal sides

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Finding Coterminal Angles

Add multiples of 360˚ Subtract Multiples of 360˚Example: Find 4 coterminal angles of 60˚60˚ + 360˚ = 420˚ 60˚ + 720˚ =

780˚60˚ – 360˚ = -300˚ 60˚ – 720˚ = -

660˚

Answer: 420˚, 780˚, -300˚, -660˚

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Defining Angles

Right Angles measure 90˚

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Finding Complimentary Angles

For degrees: = 90˚ - θor = 89˚ 59’ 60” – θ

Example: Find the angle complementary to 73.26˚

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Finding Complementary AnglesExample 2: Find the angle that is

complementary to 25˚ 43’ 37”.

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Finding Complementary Angles For Radians

= π/2 – θExample: Find the complementary angle of

π/4 radians.

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Finding Supplementary Angles For degrees

= 180˚ - θ For radians

= π - θ

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Converting Between Radians and Degrees

To Change Multiply by Example

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Converting Decimal Degrees to Degrees/Minutes/Seconds

D˚ M’ S” = D˚ + ˚ + ˚

Example: Convert 19˚ 47’ 23” to decimal degrees.

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3600

S

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Converting Radians to Degrees/Minutes/Seconds

Convert radians to decimal degrees Non-decimal portion is in degrees

Multiply decimal portion by 60’ Non-decimal portion is minutes

Multiply decimal portion by 60” & round Seconds

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Converting Radians to Degrees/Minutes/Seconds (Cont.)

Example: If θ =3 radians, approximate θ in terms of degrees/minutes/seconds.