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Chapter 4: Circular Functions

Chapter 4: Circular Functions. Lesson 1: Measures of Angles and Rotations Mrs. Parziale. Do Now. Given a radius of 1 for the circle to the right, find the following in terms of pi ( ) The circumference of the circle. The length of a 180 ° arc. The length of a 90 ° arc.

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Chapter 4: Circular Functions

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  1. Chapter 4: Circular Functions Lesson 1: Measures of Angles and Rotations Mrs. Parziale

  2. Do Now • Given a radius of 1 for the circle to the right, find the following in terms of pi () • The circumference of the circle. • The length of a 180° arc. • The length of a 90° arc. • The length of a 45° arc. 1

  3. Terms To Know • angle – the union of two rays with a common endpoint. • sides – are examples • vertex– The point at which the two rays meet. B is the vertex in this example.

  4. More Terms to Know • rotation image - is the rotation image of about the vertex B • counterclockwise rotations – are positive. • clockwise rotations – are negative. Measure of an angle represents its size and direction.

  5. Revolutions • Rotations can be measured in revolutions. • 1 counterclockwise revolution = 360° To convert To convert revolutions to degrees to degrees: revolutions:

  6. Example 1: (a) revolution counterclockwise (b) revolution clockwise (c) revolution clockwise (d) revolutions counterclockwise

  7. Radians • Radians have only been around for about 100 years. • Radians are another means of measuring angle based on how far it travels on the unit circle. • Primary use of radians was to simplify calculationsusing angle measures. • Relate to the circumference of a circle with radius of 1

  8. More Radians • circumference of a circle • circumference of a circle with radius of 1 • With one revolution of a circle

  9. Example 2: 1 rev = 2 • Convert to radians. Give exact values (in terms of pi): • revolution counterclockwise • revolution clockwise • revolution counterclockwise • revolution clockwise

  10. Example 2, cont: • Convert to radians. Give exact values: (e) revolution clockwise (f) revolution counterclockwise

  11. Unit Circle • Converting radians to degrees: • Converting degrees to radians:

  12. Unit Circle How many radians in 30°? How many degrees in ?

  13. Example 3: • Convert to radians. Give both exact and approximate values (hundredth):

  14. Example 4: • How many revolutions equal 8 radians (approx)? (Set up a proportion.)

  15. Example 5: • How many revolutions equal radians (exact?)

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