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Chapter 5 Trigonometric Functions

Chapter 5 Trigonometric Functions. Section 5.1 Angles and Degree Measure. Standard Position. An angle is in standard position when its vertex is at the origin and its initial side is along the positive x-axis. Positive and Negative Angles.

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Chapter 5 Trigonometric Functions

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  1. Chapter 5Trigonometric Functions Section 5.1 Angles and Degree Measure

  2. Standard Position • An angle is in standard position when its vertex is at the origin and its initial side is along the positive x-axis.

  3. Positive and Negative Angles • If you rotate in a counter-clockwise direction, the angle formed is POSITIVE • If you rotate in a clockwise direction, the angle formed is NEGATIVE

  4. Positive and Negative Angles

  5. Examples of Angles in Std Position

  6. Degrees/ Minutes/ Seconds • Degree: The most common unit used to measure the size of angles • Minutes: Each degree is subdivided into 60 minutes (notation 1 minute = 1 ‘) • Seconds: Each minute is subdivided into 60 seconds (notation 1 second = 1’’)

  7. Example 1 • Change the following measurements into degrees, minutes and seconds. • 329.125° =329 ° + (.125 ⋅ 60) =329 ° + 7.5’ =329 ° + 7’ + (.5 ⋅ 60)’’ =329 °7’30’’ • 35.202 ° =35 °12’7’’

  8. Example 2 • Write the following angles in decimals rounded to the nearest thousandth • 34°57’ = 34° + 57 (1/60) =34.95° b. 72 °46’30’’ =72°+ 46 (1/60) + 30(1/3600) = 72.775°

  9. Quadrantal Angles • Angles formed when the terminal side coincides with one of the axes

  10. Rotations • A full rotation around the circle is 360° • Measures of more than 360 °represent multiple rotations

  11. Example 3 • Give the angle measure represented by each rotation • 9.5 rotations clockwise (9.5)(-360 °)= -3420 ° • 6.75 rotations counter-clockwise (6.75)(360°) = 2430 °

  12. Coterminal Angles • Two angles in standard position are coterminal if they have the same terminal side. • Every angle has infinitely many coterminal angles.

  13. Coterminal Angles

  14. Example 4 • Identify 2 positive and 2 negative angles that are coterminal with each angle below • 86 ° • 294 °

  15. Example 5 • If each angle is in standard position, determine a coterminal angle that is between 0° and 360°. State the quadrant in which the terminal side lies. • 595 ° 235° (Quadrant III) b. -777 ° -57 ° or 303 ° (Quadrant IV)

  16. Reference Angle • The acute angle formed by the terminal side of the given angle and the x-axis

  17. Example 6 • Find the measure of the reference angle • 312° 48° 2. -195° 15° 3. 120° 60° 4. -135° 45°

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