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Angles and their Measure

Angles and their Measure. Geometric Representation of Angles. Definition of Angles. Angles Initial Side and Standard Position. Angles. Degrees: One degree is 1/360 of a revolution. A right angle is an angle that measures 90 degrees or ¼ revolution

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Angles and their Measure

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  1. Angles and their Measure Geometric Representation of Angles

  2. Definition of Angles • Angles • Initial Side and Standard Position

  3. Angles • Degrees: One degree is 1/360 of a revolution. • A right angle is an angle that measures 90 degrees or ¼ revolution • A straight angle is an angle that measures 180 degrees or ½ revolution

  4. Angles • Drawing an Angle • (a) 45 degrees • (b) -90 degrees • (c) 225 degrees • (d) 405 degrees

  5. Converting between Degrees, Minutes, Seconds and Decimal • 1 degree equals 60’ (minutes) • 1’ (minute) equals 60” (seconds) • Using graphing calculator to convert

  6. Radians • Definition • Arc Length • For a circle of radius r, a central angle of q radians subtends an arc whose length s is • s=r q

  7. Finding the Length of a Circle • Find the length of the arc of a circle of radius 2 meters subtended by a central angle of 0.25 radian. • s=r q with r = 2 meters and Θ = 0.25 • 2(0.25) = 0.25 meter

  8. Relationship Between Degrees and Radians • One revolution is 2π therefore, 2πr = rθ (arc length formula) • It follows then that 2π = θ and • 1 revolution = 2π radians • 360 degrees = 2π radians • or 180 degrees = π radians • so . . . 1 degree = π/180 radian and • 1 radian = 180/π degrees

  9. Converting from Degrees to Radians • Convert each angle in degrees to radians: • (a) 60 degrees • (b) 150 degrees • (c) – 45 degrees • (d) 90 degrees

  10. Converting Radians to Degrees • Convert each angle in radians to degrees • (a) π/6 radian • (b) 3π/2 radian • (c) -3π/4 • (d) 7π/3

  11. Common Angles in Degrees and Radians • Page 375 has common angles in degree and radian measures

  12. Finding Distance Between two Cities • Steps: • (1) Find the measure of the central angle between the two cities • (2) Convert angle to radians • (3) Find the arc length (remember we live on a sphere and the distance between two cities on the same latitude is actually an arc length)

  13. Area of a Sector • The area A of the sector of a circle of radius r formed by a central angle of θ radians is • A = ½ r^2θ • Examples

  14. Circular Motion • Linear Speed: • v = s/t • Angular Speed: • ω = θ/t

  15. Circular Motion • Angular Speed is usually measured in revolutions per minute (rpms). • Converting to radians per minute • Linear Speed given an Angular Speed: • v = rω • where r is the radius

  16. Finding Linear Speed • A child is spinning a rock at the end of a 2-ft rope at the rate of 180 rpms. Find the linear speed of the rock when it is released.

  17. Cable Cars of San Francisco • At the Cable Car Museum you can see four cable lines that are used to pull cable cars up and down the hills of San Francisco. Each cable travels at a speed of 9.55 miles per hour, caused by rotating wheel whose diameter is 8.5 feet. How fast is the wheel rotating? Express your answer in rpms.

  18. Circular Motion • On-line Examples • On-line Tutorial

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