We will begin our study of precalculus by focusing on the topic
of trigonometry Literal meaning of trigonometry The measurement of
triangles Thus, we will be spending a lot of time working with and
studying different triangles
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Radian and Degree Measure To begin our study on trigonometry,
we first start with angles and their measures An angle is
determined by rotating a ray (half-line) about its endpoint.
Initial Side Terminal Side Vertex
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Radian and Degree Measure Standard Position An angle in
standard position has 2 characteristics: 1) Initial side lies on
the x-axis 2) Vertex is at the origin x y
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Radian and Degree Measure Standard Position An angle in
standard position has 2 characteristics: 1) Initial side lies on
the x-axis 2) Vertex is at the origin x y
Slide 6
Radian and Degree Measure Standard Position An angle in
standard position has 2 characteristics: 1) Initial side lies on
the x-axis 2) Vertex is at the origin x y
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Radian and Degree Measure Positive Angles Rotate clockwise In
standard position, start by going up Negative Angles Rotate
counterclockwise In standard position, start by going down
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Radian and Degree Measure Negative Angle
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Radian and Degree Measure Positive Angle
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Radian and Degree Measure Angles can be measured in one of two
units: 1) Degrees 2) Radians One full revolution of a central angle
would be equal to: 1) 360 2) 2 radians (or 6.28 radians)
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Radian and Degree Measure In radians, there are common angles
that will need to be memorized = 180 = 90 = 270
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Radian and Degree Measure In addition to our quadrant angles,
there are 3 more angles that we will be using throughout the year.
= 30 = 45 = 60
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Radian and Degree Measure Coterminal Angles Two angles that
have the same: Vertex Initial Side Terminal Side All angles have an
infinite number of coterminal angles
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Radian and Degree Measure Finding Coterminal angles To find a
positive coterminal angle Add 2 (or 360 ) to the given angle To
find a negative coterminal angle Subtract 2 (or 360 ) from the
given angle
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Radian and Degree Measure Graph the following angle and
determine two coterminal angles, one positive and one
negative.
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Radian and Degree Measure Graph the following angles and find
two coterminal angles, one positive and one negative.
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Radian and Degree Measure
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Section 4.1 Radian and Degree Measure
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Graph the following angles and find two coterminal angles, one
positive and one negative.
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Radian and Degree Measure
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Yesterday we covered: Angles in degrees and radians Coterminal
angles Today we are going to cover: Complementary and supplementary
angles Converting between degrees and radians Converting minutes
& seconds to degrees
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Radian and Degree Measure Complementary Angles Two positive
angles whose sum is (or 90 ) Supplementary Angles Two positive
angles whose sum is of (180 )
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Radian and Degree Measure Find the complement and supplement to
the following angle. supplement:
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Radian and Degree Measure Find the complement and supplement of
the following angles:
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Radian and Degree Measure Conversions between degrees and
radians 1. To convert degrees to radians, multiply degrees by: 2.
To convert radians to degrees, multiply radians by:
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Radian and Degree Measure Degrees to Radians Radians to
Degrees
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Radian and Degree Measure Convert the following from degrees to
radians.
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Radian and Degree Measure Convert the following from radians to
degrees.
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Section 4.1 Radian and Degree Measure
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Find the complement, supplement, and two coterminal angles of
the following angle. Convert the angle above to degrees.
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Radian and Degree Measure So far in this section, we have: a)
Graphed angles in both radians & degrees b) Found positive and
negative coterminal angles c) Found complementary and supplementary
angles d) Converted between radians and degrees Today we are going
to apply this to different word problems (arc length, linear speed,
angular speed)
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Radian and Degree Measure Arc Length The distance along the
circumference of a circle with a central angle of Given by the
formula: s = r Where: s = the arc length r = the radius of the
circle = the central angle in radians
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Radian and Degree Measure A circle has a radius of 4 inches.
Find the length of the arc intercepted by a central angle of 240.
1) Convert the angle to radians. 2) Apply the formula. S =(4)=16.7
inches
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Radian and Degree Measure On a circle with a radius of 9
inches, find the length of the arc intercepted by a central angle
of 140 . S =(9) = 22 inches
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Radian and Degree Measure Linear & Angular Speed Linear
speed measures how fast a particle is moving along the circular arc
of a circle with radius r Given by the formula:
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Radian and Degree Measure Linear & Angular Speed Angular
speed measures how fast the angle changes Given by the
formula:
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Radian and Degree Measure The second hand of a clock is 11
inches long. Find the linear speed of the tip of this second hand
as it passes around the clock face. In one revolution, how far does
the tip travel? s = 2 r = 22 inches What is the time required to
travel this distance? t = 60 seconds r = 11 inches
Slide 41
Radian and Degree Measure The second hand of a clock is 11
inches long. Find the linear speed of the tip of this second hand
as it passes around the clock face. s = 22 inchest = 60 seconds
Linear Speed =
Slide 42
Radian and Degree Measure A car is moving at a rate of 65 mph,
and the diameters of its wheels is 2.5 feet. a) Find the number of
revolutions per minute the wheels are rotating. b) Find the angular
speed of the wheels in radians per minute.
Slide 43
Radian and Degree Measure A car is moving at a rate of 65 mph,
and the diameters of its wheels is 2.5 feet. a) Find the number of
revolutions per minute the wheels are rotating. Find the arc length
for one revolution: S = r = (1.25) (2 ) = 2.5 feet per revolution
How many feet per hour is the car traveling? (65 mph )(5,280 feet)=
343,200 feet/hour = 5,720 feet/min = 728.3 revolutions
Slide 44
Radian and Degree Measure A car is moving at a rate of 65 mph,
and the diameters of its wheels is 2.5 feet. b) Find the angular
speed of the wheels in radians per minute. (728.3 revolutions) (2 )
= 4,576 radians Angular Speed
Slide 45
Radian and Degree Measure A car is moving at a rate of 35 mph,
and the radius of its wheels is 2 feet. a) Find the number of
revolutions per minute the wheels are rotating. b) Find the angular
speed of the wheels in radians per minute. 245.1 revolutions per
minute