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Understanding Angle Measurement: Degrees and Radians

This lesson focuses on measuring angles using degrees and radians, highlighting the importance of converting between these two units. Students will learn to write formulas for arc length, understand the historical context of angle measurement, and solve problems involving radians and degrees. Key objectives include measuring angles in radians, converting between degrees and radians, and finding trigonometric function values in radian measure. Through examples and classwork, students will gain a comprehensive understanding of angle measurement in precalculus.

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Understanding Angle Measurement: Degrees and Radians

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  1. GBK PrecalculusJordan Johnson

  2. Today’s agenda • Greetings • Intro • Lesson: Angle Measure • Classwork / Homework • Clean-up

  3. Important numbers • 0, 6, 4, 3, 2.

  4. Angles & Arc Length: Problem • Write a formula for the length of an arc, subtended by an angle of °, with radius r.

  5. Angles & Arc Length: Answer • Write a formula for the length of an arc, subtended by an angle of °, with radius r.

  6. Problem: • Ugliness: • Extra factor of 1/360. • Formula requires a fraction. • Goal: Eliminate that fraction. • Historical trivia: • Division into 360° is believed to come from approximate year length. • Division of degrees into minutes/seconds is from Babylonian sexagesimal number system. • Proposed solution that never caught on: divide a right angle into 100 units called gradians (grad). • Still requires the fraction, though.

  7. Solution: Radian Measure • We’ll solve the problem by introducing a new way of measuring angles. • Objectives: • Measure angles in radian units. • Convert between degrees and radians. • Find trig function values of radians.

  8. Definition • The radian measure of an angle  is the ratio of an arc subtended by  to the arc’s radius. • Notice: a full circle is a 360° arc, so: • Radian measures are usually given in terms of .

  9. Alternate Definition • Equivalently, the radian measure of an angle  is the length of an arc on the unit circle subtended by ’s radii.

  10. Examples  Length of arc if r = 1 • /4 • /3 • 5 • 7 • /4 • /3 • 5 • 7

  11. Examples  Length of arc if r = 8 • /4 • /3 • 5 • 7 • (/4)8 = 8/4 = 2 • (/3)8 = 8/3 • 5  8 = 40 • 7  8 = 56

  12. Degrees  Radians • Converting from degrees to radians: • Given  in degrees, • Example: • Convert 80° to radians.Give both exact & approximate answers. • Answer: Exactly 4/9. (Approximately 1.3962634.)

  13. Radians  Degrees • Converting from radians to degrees: • Given  in radians, • Example: • Convert 12.5664 radians to degrees.Give approximate answer (to nearest integer). • Answer: About 720°. (Note that 12.5664  4.)

  14. Important numbers 0 6 4 3 2  0 /6 /4 /3 /2 Convert to degrees:  0° 30° 45° 60° 90°

  15. Homework • From Section 3-4: • Read pp. 110-115 and answer the Reading Analysis questions. • Do exercises Q1-Q10. • Do #1-3, 9, 11, 17, 21, 25, 29, 31, and 37-53 odd. • Due Tuesday, 11/2/2010.

  16. Clean-up / Reminders • Pick up all trash / items. • Push in chairs (at front and side tables). • See you tomorrow!

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