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Lesson 4.1 Angles and Angle Measure
1
When drawing an angle on the xy axis in standard position, the following conditions must apply:
• its vertex is at the _________
• its initial arm lies along the ________________
(A) Standard Position
CHAPTER 4
Lesson 4.1: Angles and Angle Measure
Trigonometry andThe Unit Circle
Lesson 4.1 Angles and Angle Measure
2
(B) Positive and Negative Rotation (Standard Position)
Positive angles:
counterclockwise rotation
Negative angles:
clockwise rotation
(C) Reference Angleangle between the terminal arm and the xaxis
Lesson 4.1 Angles and Angle Measure
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Angle Measure
There are two units for measuring angles: » degrees» radians
One degree is defined as of a full rotation
is the measure of the angle formed when the arc length of a circle has the same length as the radius.
One radian
Lesson 4.1 Angles and Angle Measure
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Example 1: What if the arclength and the radius of the circle are equal?
What is the measure of 1 radian in degrees?
http://www.mathsisfun.com/geometry/radians.html
Lesson 4.1 Angles and Angle Measure
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Example 2: What are some common radian and degree measures?
Lesson 4.1 Angles and Angle Measure
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Convert From Radians to Degrees and vice versa
Example 3: Convert the following to radian measure (exact and approximate).
(i) (ii)
(iii) (iv)
(v)
1 radian =
Formula:
To convert from radians to degrees To convert from degrees to radians
1 degree =
Lesson 4.1 Angles and Angle Measure
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Example 4: Convert the following to degree measure.
(i) (ii)
(iii) (iv)
(v)
Lesson 4.1 Angles and Angle Measure
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Example 5: Sketch the following angles in standard position.
(i) (ii)
(iii) (iv)
(v) 2.57
P175 #1, 3, 4, 5abd, 6
Lesson 4.1 Angles and Angle Measure
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Coterminal Angles
Example 1: Show that 600 and 3000 are coterminal angles.
Note: Coterminal anges can be found by adding or subtracting multiples of 3600 or 2
angles in standard position with the same terminal arm and can be measured in degrees or radians
Lesson 4.1 Angles and Angle Measure
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Example 2: Name one positive and one negative angle measure that is coterminal with each angle.
(ii) 3100
(iii) 5150 (iv)
(i) 400
Lesson 4.1 Angles and Angle Measure
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Your Turn: Name one positive and one negative angle measure that is coterminal with each angle.
(i) 2700 (ii) 7400
(iii) (iv)
Lesson 4.1 Angles and Angle Measure
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General Form:
Coterminal Angles in General Form
• Is it possible to list all of the possible coterminal angles?
• It is possible to write a formula to represent the group of angles in our solution.
Example 1:
Coterminal angles in CCW (+) direction Coterminal angles in CW () direction
Combine the solutions in both directions to list ALL possible coterminal angles (in degree and radian measure)
Lesson 4.1 Angles and Angle Measure
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Example 2:
(i) Express the angles coterminal with 110o in general form.
(ii) Identify the angles coterminal with 110o that satisfy the domain
Example 3:
(i) Write an expression for all possible angles coterminal with .
(ii) Identify the angles that are coterminal that satisfy .
P.176 #7, 8, 9, 11abdf
Lesson 4.1 Angles and Angle Measure
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Arc Length of a Circle
Determine a formula relating the radius (r), central angle and arc length of a circle ( ).
know how to rearrange!
oA
B
oA
B
minor arc/major arc
measuredin radians
, r measured in same units
Lesson 4.1 Angles and Angle Measure
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Example 1: Determine the measures of the arc length subtended by the angles and radii below:
(i) central angle of with radius of 10cm.
(ii) central angle of 2.6 rad with radius of 4.9cm.
Example 2: Determine the measure of the radius of a circle in the following diagram.
Lesson 4.1 Angles and Angle Measure
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Example 3:
Determine the missing quantities where represents the length of the arc of a circle with radius r, subtended by a central angle of .
(i) r =8.7cm, =750, = ? cm
(ii) r = ? cm, =1.8, = 4.7 mm
(iii) r = 5m, =?, = 13 m
P.176 #12ac, 13
Lesson 4.1 Angles and Angle Measure
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Coterminal angles angles in standard position with the same terminal arm and can be measured in degrees or radians.
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