Transcript Angles, Degrees, and Special Triangles

Radian & Degree Measure
MATH 109 - Precalculus
S. 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.   QIII means θ is in standard position
with its terminal side in Quadrant III
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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 revolution
90° 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
s where θ is in radians
 
r
– i.e. How many radii r comprise the arc length s
• For θ = 1 radian,
s=r
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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
s 2r
  
 2
revolution
r
r
• Therefore, θ = 360° = 2π consists of a complete
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revolution around a circle
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  rad
so that degrees will divide out
180 
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 180 
so that radians will divide out  rad
leaving degrees
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Common Angles
• Need to become familiar with the degree and
Deg
Rad
radian conversion
0°
0
between the following
30°
⁄
commonly used angle
45°
⁄
measurements:
π
π
6
4
60°
π⁄
3
90°
π⁄
2
180°
π
270°
3π⁄
2
360°
2π
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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)  

7
c)   115 
b)   4.2
d)   532 
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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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