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4.1: Radian and Degree Measure

4.1: Radian and Degree Measure. Objectives: To use radian measure of an angle To convert angle measures back and forth between radians and degrees To find coterminal angles To find arc length, linear speed, and angular speed. We are going to look at angles on the coordinate plane….

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4.1: Radian and Degree Measure

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  1. 4.1: Radian and Degree Measure Objectives: To use radian measure of an angle To convert angle measures back and forth between radians and degrees To find coterminal angles To find arc length, linear speed, and angular speed

  2. We are going to look at angles on the coordinate plane… • An angle is determined by rotating a ray about its endpoint • Starting position: Initial side (does not move) • Ending position: Terminal side (side that rotates) • Standard Position: vertex at the origin; initial side coincides with the positive x-axis • Positive Angle: rotates counterclockwise (CCW) • Negative Angle: rotates clockwise (CW)

  3. Negative Angle Positive Angles

  4. 1 complete rotation: 360⁰ • Angles are labeled with Greek letters: α (alpha), β (beta), and θ (theta) • Angles that have the same initial and terminal sides are called coterminal angles

  5. RADIAN MEASURE (just another unit of measure!) • Two ways to measure an angle: radians and degrees • For radians, use the central angle of a circle • s= arc length intercepted by angle • One radian is the measure of a central angle, Ѳ, that intercepts an arc, s, equal to the length of the radius, r r s=r

  6. One complete rotation of a circle = 360° • Circumference of a circle: 2 r • The arc of a full circle = circumference s= 2 r Since s= r, one full rotation in radians= 2 =360 ° , so just over 6 radians in a circle

  7. (1 revolution) ½ a revolution = ¼ a revolution 1/6 a revolution= 1/8 a revolution=

  8. Quadrant 2 Quadrant 1 Quadrant 3 Quadrant 4

  9. Coterminal angles: same initial side and terminal side Name a negative coterminal angle:

  10. You can find an angle that is coterminal to a given angle by adding or subtracting Find a positive and negative coterminal angle:

  11. Finding Complementary and Supplementary Angles Complement for : Supplement for : If >90, it has no complement. If > 180, it has no supplement.

  12. Find the complement and the supplement of the angle, if possible.

  13. Degree Measure So……… Converting between degrees and radians: • Degrees →radians: multiply degrees by • Radians → degrees: multiply radians by

  14. Convert to Radians: • 320° • 45 ° • -135 ° • 270 ° • 540 °

  15. Convert to Radians:

  16. Sketching Angles in Standard Position: Vertex is at origin, start at 0° 1. 2. 60°

  17. 3.

  18. Finding Arc Length: arc length = s length of radius = r central angle = (in radians) and

  19. Examples: • A circle has a radius of 4 inches. Find the length of the arc intercepted by and angle of 120°. • Find if the arc length is 32m and r = 7 m.

  20. Linear and Angular Speed The formula for the length of a circular arc can be used to analyze the motion of a particle moving at a constant speed along a circular path. Linear speed: (how fast the particle moves) Angular speed: (how fast the angle changes)

  21. A neighborhood carnival has a Ferris Wheel whose radius is 30 ft. You measure the time it takes for one revolution to be 70 sec. What is the linear and angular speed of the Ferris Wheel?

  22. 2. A lawn roller with a 10-inch radius makes 1.2 revolutions per second. a.) Find the angular speed per second. b.) Find the speed of the tractor pulling the roller.

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