Angles & Angle Measures 33 22 11 Notation, Definitions& Measurement of Angles, Coterminal, Right, Complementary, Supplementary Angles & Intro to Radians.

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Presentation transcript:

Angles & Angle Measures Notation, Definitions& Measurement of Angles, Coterminal, Right, Complementary, Supplementary Angles & Intro to Radians Practice Problems

Notation  Variables for angles  Frequently Greek letters  α (alpha)  β (beta)  γ (gamma)  θ (theta) 2

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 3

Measurement  Clockwise (CW)  Negative Angle  Counter-Clockwise (CCW)  Positive Angle 4

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” 5

Measurement (Cont.)  Radians  Similar to degrees  Always measured in terms of pi ( π ) 360˚/0˚ = 2 π 90˚ = π /2 180˚ = π 270˚ = 3 π /2 6

Coterminal Angles  Have the same initial and terminal sides 7

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˚ 8

Defining Angles  Right Angles measure 90˚ 9

Finding Complimentary Angles  For degrees:  = 90˚ - θ or  = 89˚ 59’ 60” – θ Example: Find the angle complementary to 73.26˚ 10

Finding Complementary Angles Example 2: Find the angle that is complementary to 25˚ 43’ 37”. 11

Finding Complementary Angles  For Radians  = π /2 – θ Example: Find the complementary angle of π /4 radians. 12

Finding Supplementary Angles  For degrees  = 180˚ - θ  For radians  = π - θ 13

Converting Between Radians and Degrees To ChangeMultiply byExample 14

Converting Decimal Degrees to Degrees/Minutes/Seconds D˚ M’ S” = D˚ + ˚ + ˚ Example: Convert 19˚ 47’ 23” to decimal degrees. 15

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 16

Converting Radians to Degrees/Minutes/Seconds (Cont.) Example: If θ =3 radians, approximate θ in terms of degrees/minutes/seconds. 17