Setion 4.1 Angle and Angle Measure - Prince Edward Island€¦ · Find the indicated angle or ......
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Section 4.1.notebook 1 October 13, 2014 Setion 4.1 Angle and Angle Measure
Transcript of Setion 4.1 Angle and Angle Measure - Prince Edward Island€¦ · Find the indicated angle or ......
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Setion 4.1
Angle and Angle Measure
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An angle is in standard position when its vertex is at the origin and its initial ray or arm is on the positive xaxis. The other ray is called the terminal arm.
If the angle of rotation is counterclockwise, the angle is positive. If the angle of rotation is clockwise, the angle is negative.
Angle in standard position
Initial arm
terminal arm
reference angle (always with the xaxis)
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Positive angle Counter clockwise rotation
Negative angle Clockwise rotation
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Note: In Math521B you would ususally see
This would indicate that we were looking for an answer within one rotation. Now we will start looking for answers within two, three or more rotations.
Two positive rotations.
Three positive rotations.
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We are familiar with angles being measured in degrees. A whole circle is measures 3600.
00,3600
900
1800
2700
In this chapter we will learn how to measure angle with radians. You can think of it as distance can be measured in miles or kilometers.
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Arc length is given by the formula:
A radian (rad) is defined as the angle formed by rotating the radius of a circle through an arc length equal to the radius.
θ must be expressed in radians
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To convert between radians and degrees, think of the circumference of a circle:To convert between radians and degrees, think of the circumference of a circle:To convert between radians and degrees, think of the circumference of a circle:To convert between radians and degrees, think of the circumference of a circle:
arc length is the entire circle
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00,3600
900
1800
2700
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There are different ways to remember how to convert between radians and degrees and viceversa.
Knowing that 1 hour = 3600 s and 1 km=1000m convert 90km/h to m/s.
If 1 mol = 6.02 x 1023 molecules how many moles is 9.72 x 1039 molecules?
So if convert 5.2 radians to degrees.
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To change from degrees to radians, multiply the measure in degrees by
Convert to radians:
To change from radians to degrees, multiply the measure in radians by
Convert to degrees:
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4.1
Example 1: Your TurnDraw each angle in standard position. Change each degree measure toradians and each radian measure to degrees. Give answers as both exactand approximate measures (if necessary) to the nearest hundredth ofa unit.a) –270° b) 150°c) d) –1.2
Answer
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Note: When an angle is given without a degree sign, it is assumed to be in radians.
Find the indicated angle or radius:
r = 4 cm
15 cm
80o
12 m
r
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Angles that have the same initial and terminal arm are called coterminal angles. The least possible positive coterminal angle is called the principal angle. or
Find the principal angle of 400.
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Find coterminal angles in general form
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What is the principal angle of 4000?
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What is the principal angle for the following?
a. 10000 b. 4250 c.
d. e. f. 9.24 rads
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4.1
Example 2: Your TurnFor each angle in standard position, determine one positive andone negative angle measure that is coterminal with it.a) 270° b) c) 740°
Answer
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4.1Example 3
Continue Next Page
Express Coterminal Angles in General Forma) Express the angles coterminal with 110° in general form. Identify the angles coterminal with 110° that satisfy the domain –720° ≤ θ < 720°.
the angles coterminal with in the domain –4π ≤ θ < 4π.
b) Express the angles coterminal with in general form. Identify
a) Angles coterminal with 110° occur at 110° ± (360°)n, n ∈ N. Substitute values for n to determine these angles.
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n 1 2 3
110°– (360°)n
110°+ (360°)n
From the table, the values that satisfy the domain –720° ≤ θ < 720°are –610°, –250°, and 470°. These angles are coterminal.
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2
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4.1 Example 3 Continued
Express Coterminal Angles in General Form
n 1 2 3 45
b) ± 2πn, n ∈ N, represents all angles coterminal with .
Substitute values for n to determine these angles.
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The angles in the domain –4π ≤ θ < 4π that are coterminal are 6
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4.1
Example 3: Your Turn
Write an expression for all possible angles coterminal with eachgiven angle. Identify the angles that are coterminal that satisfy–360° ≤ θ < 360° or –2π ≤ θ < 2π.a) –500° b) 650° c)
Answer
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To change from degrees to radians, multiply the measure in degrees by
Convert to radians:
To change from radians to degrees, multiply the measure in radians by
Convert to degrees:
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Note: When an angle is given without a degree sign, it is assumed to be in radians.
Find the indicated angle or radius:
r = 4 cm
15 cm
80o
12 m
r
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Angles that have the same initial and terminal arm are called coterminal angles. The least possible positive coterminal angle is called the principal angle.
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