Notes ­ October 19
Objective: Demonstrate an understanding of angles in standard position, expressed in degrees and radians. Match the angle with its measure (in degrees). a) 700 b) 1350 c) 3000 d) ­700 e) 2200 1
Notes ­ October 19
Match the angle with its measure (in degrees). a) 4300 b) ­2250 c) 10200 d) ­7900 e) 5800 coterminal angles: angles in standard position with the same terminal arms
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Notes ­ October 19
Definition: One radian is the measure of the central angle subtended in a circle by an arc equal in length to the radius of the circle. degrees
radians
one full rotation: one half rotation: one quarter rotation: one eighth rotation:
Note: Angle measures without units are assumed to be in radians. Example: For each angle in standard position, determine one positive and one negative angle measure that is coterminal with it. a) 2700
b) ­5π/4
c) 7400
The General Form
Any given angle has an infinite number of angles coterminal with it. Angles coterminal with any angle can be expressed as: 3
Notes ­ October 19
Unit Conversions
Given any angle θ, we can use ratios to convert from degree measure to radian measure, and from radian measure to degree measure. 1 full rotation = degrees 1 full rotation = radians
Therefore, 3600 = 2π radians
So, to convert from radians to degrees, we can multiply by
OR
and to convert from degrees to radians we can multiply by
OR
Example: Change each degree measure to radian measure and each radian measure to degree measure. a) 300
b) ­1200
c) 5π/4
d) 2.57
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Notes ­ October 19
Arc Length of a Circle
a = r
Caution: This formula works for any circle, provided that is measured in radians, and a and r are both measured in the same units. 5
Notes ­ October 19
Example
A circle with a radius of 16.2 cm is drawn on a large piece of cardboard. A central angle of 74O is drawn. What is the length of the arc subtended by this angle, rounded to the nearest tenth of a cm? Example
The wiper on the rear window of a particular car moves through an angle of 5/6 rad in a single sweep and clears a region that is 16in wide, as shown in the diagram. Find the total perimeter of the cleared region, to the nearest inch. Summary: • angles can be measured using different units, including degrees and radians
• an angle measured in one unit can be converted to the other using the relationships 1 full rotation = 3600 = 2π
• angles that are coterminal have the same initial arm and the same terminal arm
• an angle has an infinite number of coterminal angles
Homework
page 175­179 # 3, 4, 6, 9, 11, 12, 14, 19, C1
Push Yourself! Try #27
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