transcript
- Slide 1
- Section 4.1 Radian and Degree Measure
- Slide 2
- 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
- Slide 3
- 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
- Slide 4
- 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 5
- 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
- Slide 7
- 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
- Slide 8
- Radian and Degree Measure Negative Angle
- Slide 9
- Radian and Degree Measure Positive Angle
- Slide 10
- 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)
- Slide 11
- Radian and Degree Measure In radians, there are common angles
that will need to be memorized = 180 = 90 = 270
- Slide 12
- 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
- Slide 13
- 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
- Slide 14
- 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
- Slide 15
- Radian and Degree Measure Graph the following angle and
determine two coterminal angles, one positive and one
negative.
- Slide 16
- Radian and Degree Measure Graph the following angles and find
two coterminal angles, one positive and one negative.
- Slide 17
- Radian and Degree Measure
- Slide 18
- Slide 19
- Slide 20
- Slide 21
- Section 4.1 Radian and Degree Measure
- Slide 22
- Graph the following angles and find two coterminal angles, one
positive and one negative.
- Slide 23
- Radian and Degree Measure
- Slide 24
- 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
- Slide 25
- Radian and Degree Measure Complementary Angles Two positive
angles whose sum is (or 90 ) Supplementary Angles Two positive
angles whose sum is of (180 )
- Slide 26
- Radian and Degree Measure Find the complement and supplement to
the following angle. supplement:
- Slide 27
- Radian and Degree Measure Find the complement and supplement of
the following angles:
- Slide 28
- 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:
- Slide 29
- Radian and Degree Measure Degrees to Radians Radians to
Degrees
- Slide 30
- Radian and Degree Measure Convert the following from degrees to
radians.
- Slide 31
- Radian and Degree Measure Convert the following from radians to
degrees.
- Slide 32
- Section 4.1 Radian and Degree Measure
- Slide 33
- Find the complement, supplement, and two coterminal angles of
the following angle. Convert the angle above to degrees.
- Slide 34
- 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)
- Slide 35
- 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
- Slide 36
- 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
- Slide 37
- 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
- Slide 38
- 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:
- Slide 39
- Radian and Degree Measure Linear & Angular Speed Angular
speed measures how fast the angle changes Given by the
formula:
- Slide 40
- 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
- Slide 46
- Radian and Degree Measure