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

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Do Now Given a radius of 1 for the circle to the right, find the following in terms of pi ( ) 1.The circumference of the circle. 2.The length of a 180 ° arc. 3.The length of a 90 ° arc. 4.The length of a 45 ° arc. 1

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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.

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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.

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

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

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Radians Radians have only been around for about 100 years. Radians are another means of measuring angle sides. Primary use of radians was to simplify calculations using angle measures. Relates to the circumference of a circle with radius of 1

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

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Example 2: Convert to radians. Give exact values (in terms of pi): a) revolution counterclockwise b) revolution clockwise c) revolution counterclockwise d) revolution clockwise 1 rev = 2

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

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Unit Circle Converting radians to degrees: Converting degrees to radians:

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

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

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

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

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4-1. Thinking about angles differently: Rotating a ray to create an angle Initial side - where we start Terminal side - where we stop.

4-1. Thinking about angles differently: Rotating a ray to create an angle Initial side - where we start Terminal side - where we stop.

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