radian & degree measure math 109 - precalculus s. rook
TRANSCRIPT
Radian & Degree Measure
MATH 109 - PrecalculusS. Rook
Overview
• Section 4.1 in the textbook:– Angles– Degree measure– Radian measure– Converting between degrees & radians
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Angles
Angles
• Angle: describes the “space” between two rays that are joined at a common endpoint– Recall from Geometry that a ray has one
terminating side and one non-terminating side
• Can also think about an angle as a rotation about the common endpoint– Start at OA (Initial side)– End at OB (Terminal side)
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Angles (Continued)
• If the initial side is rotated counter-clockwise
θ is a positive angle
• If the initial side is rotated clockwise
θ is a negative angle
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Angles in Standard Position
• An angle θ is in standard position if its:– Initial side extends along the positive x-axis
in reference to the Cartesian Plane– Vertex is (0, 0)
• The “element of” symbol can be used to denote an angle in standard position– e.g. means θ is in standard position
with its terminal side in Quadrant III
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QIII
Degree Measure
Angle Measure
• Angle Measure: expresses the size of an angle– i.e. the space in between the initial and terminal
sides in the direction of rotation
• Two common types of angle measures:– Degrees– Radians
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Degree Measure
• 1 degree corresponds to (1⁄360) of a complete revolution starting from the initial side of an angle to its terminal side– i.e. Can be viewed in terms of a circle
• Common degree measurements to be familiar with: 360° makes one complete revolution• The initial and terminal sides of the angle are the same
180° makes one half of a complete revolution90° makes one quarter of a complete revolution
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Degree Measure (Continued)
• Angles that measure:– Between 0° and 90° are known as acute angles– Exactly 90° are known as right angles• Denoted by a small square between the initial and
terminal sides– Between 90° and 180° are known as obtuse angles
• Complementary angles: two angles whose measures sum to 90°
• Supplementary angles: two angles whose measures sum to 180°
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Degrees & Minutes• Degrees can be broken down even further using minutes
1° = 60’• To convert from decimal degrees to degrees and minutes:
– Use the decimal portion of the angle– Multiply by the appropriate conversion ratio
• Align the units in the ratio so the degrees will divide out, leaving the minutes
• To convert from degrees and minutes to decimal degrees:– Use the minutes from the angle measurement– Multiply by the appropriate conversion ratio
• Align the units in the ratio so the minutes will divide out, leaving the degrees
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Sketching Angles in Standard Position (Example)
Ex 1: Sketch each angle in standard position:
a) 293°
b) -115°
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Complementary & Supplementary Angles (Example)
Ex 2: Find: i) the complement ii) the supplement
θ = 65°
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Converting from Degrees to Minutes & Vice Versa (Example)
Ex 3: Convert a) to degrees and minutes and convert b) to decimal degrees – approximate if necessary:
a) θ = 232.55°
b) θ = 17° 22’
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Radian Measure
Motivation for Introducing Radians
• In some calculations, we require the measure of an angle (θ) to be a real number – we need a unit other than degrees– This unit is known as the radian
• Many calculations tend to become easier to perform when θ is in radians– Further, some calculations can be performed or even
simplified ONLY if θ is in radians– However, degrees are still in use in many applications so
a knowledge of both degrees and radians is ESSENTIAL
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Radians
• Radian Measure: A circle with central angle θ and radius r which cuts off an arc of length s has a central angle measure of where θ is in radians
– i.e. How many radii r comprise the arc length s
• For θ = 1 radian, s = r
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r
s
Radian Measure (Example)
Ex 4: Find the radian measure of the central angle of a circle of radius r that subtends an arc length of s
A radius of 27 inches and an arc length of 6 inches
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Converting Between Degrees and Radians
Relationship Between Degrees and Radians
• Given a circle with radius r, what arc length s is required to make one complete revolution?– Recall that the circumference measures the distance or
length around a circle– What is the circumference of a circle with radius r?
C = 2πr
• Thus, s = 2πr is the arc length of one revolution and is the number of radians in one
revolution• Therefore, θ = 360° = 2π consists of a complete revolution
around a circle20
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r
r
r
s
Relationship Between Degrees and Radians (Continued)
• Equivalently: 180° = π radians– You MUST memorize this conversion!!!
• Technically, when measured in radians, θ is unitless, but we sometimes append “radians” to it to differentiate radians from degrees– Like radians, real numbers are unitless as well
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Converting from Degrees to Radians & Vice Versa
• To convert from degrees to radians:– Multiply by the conversion ratio
so that degrees will divide out leaving radians
– If an exact answer is desired, leave π in the final answer– If an approximate answer is desired, use a calculator to
estimate π• To convert from radians to degrees:– Multiply by the conversion ratio
so that radians will divide out leaving degrees
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180
rad
rad
180
Common Angles
• Need to become familiar with the degree and radian conversion between the following commonly used angle measurements:
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Deg Rad
0° 0
30° π⁄6
45° π⁄4
60° π⁄3
90° π⁄2
180° π
270° 3π⁄2
360° 2π
Converting from Degrees to Radians & Vice Versa (Example)
Ex 5: Convert a) & b) to degrees and convert c) & d) to radians – leave π in the answer when necessary:
a) b)
c) d)
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2.4
115 532
Coterminal Angles
• Two angles are coterminal if:– BOTH are standard angles– Share the SAME terminal side
• How can we obtain an angle coterminal to an angle θ?– The second angle must terminate where θ
terminates– Recall that one complete revolution around a
circle is 360° in degrees or 2π in radians
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Coterminal Angles (Example)
Ex 6: Do the following:
a) Given θ = -190° find in degrees: i) two coterminal angles and ii) all angles coterminal to θ
b) Given θ = π⁄8 find in radians: i) two coterminal angles and ii) all angles coterminal to θ
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Summary
• After studying these slides, you should be able to:– Draw an angle in standard position– Find both the complement and supplement of an angle– Convert between degrees & minutes and decimal degrees and vice
versa– Calculate the radian measure of a circle with radius r and
subtended by an arc length s– Convert between radians & degrees and vice versa
• Additional Practice– See the list of suggested problems for 4.1
• Next lesson– Trigonometric Functions: The Unit Circle (Section 4.2)
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