Transcript
March 15, 2017
Section 4.1 Radian and Degree Measure
Trigonometry is the study of the relationships between the measures of the sides and angles of a triangle. The trigonometric relationships will be viewed as functions and the domain will be extended to all real numbers. Trig functions can represent waves(sound, light), cyclical phenomena, rotations, orbits, oscillations, etc.
An angle with measure one radian intercepts an arc whose length is equal to the radius. The radian measure of an angle tells you how many radii fit into the arc intercepted by the central angle.
radian measure of an angle = arc length radius No units on radian measure. It is a ratio and the units cancel out.
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An angle is determined by rotating a ray about its endpoint.
Standard position for an angle is for the initial side to lie on the positive x-axis with vertex at the origin. terminal side vertex
initial side
If the rotation is counterclockwise, the angle measure is positive. If the rotation is clockwise, the angle measure is negative. We have measured angles in degrees, now we will measure them in radians. Geometer's sketchpad activity.
radian measure of 1 full rotation
so 2
radians = 360
One radian is equivalent to how many degrees?
When an angle measure is given in radians, there is no symbol used.
cos 2.5 radians
sin 30°
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Rotation Radian measure of angle
Degree measure
1 1/2
2∏ ∏
360° 180°
1/4 1/6
∏/2 ∏/3
90° 60°
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Estimate the radian measure of each angle to the nearest half radian.
-2.5
3 ≈3 -5
1
In which quadrant is the terminal side of each angle?
3
3rd 2nd
2
2nd 4th 3rd
-3.5
2nd
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Converting degrees to radians.
180 =
radians
Multiply by the conversion factor radians on top, degrees will cancel out
Radians
Degrees
Converting radians to degrees. Multiply by the conversion factor radians on top, degrees will cancel out
Degrees
Radians 1.)
or just replace
2.)
1.3
with 180
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Converting between radians and degrees.
180 =
radians Degrees
Radians
4
2 3
Find, if possible, the complement and supplement of each angle. complementary
none
angle must be < 90 to have a complement.
none
supplementary
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coterminal angles have the same initial and terminal sides Find 2 coterminal angles for
just go one full rotation(2 ) in either direction from the terminal side of
Find 2 coterminal angles.
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Find the radian measure of each angle. 1.)
37.7 inches
16 in.
2.)
10 in.
5 in.
Find the length of the arc given the radius and measure of the central angle. 1.) radius = 4 ft central angle = 2 radians
2.) radius = 12 in.
central angle = ∏/6
Find the radius, given the arc length and measure of the angle. 1.) arc length = 20 in. central angle = 4 radians
2.) arc length = 8∏ ft
central angle = ∏/4
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Find the area of the sector.
Areas of sectors
If 0 is in degrees area of sector = If 0 is in radians area of sector =
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Find the area of each sector. Show work.
radians
9 in
2 in
Arc length 6 in
If in degrees:
arc length =
If
arc length =
in radians
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Find the length of each arc.
33 in
18 in
If a sector in a circle with a central angle of ∏/3 has an area of 4∏ sq meters, what is the radius of the circle? State your answer as a decimal accurate to thousandths.
meters
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A bicycle wheel with diameter 24 inches rotates at a rate of 3 revolutions per second. a.) How fast is the bicycle moving? b.) How fast is the wheel rotating in radians per second? total distance
linear speed =
total time
=
3(24 ) 1 second
= 72
inches/sec
Linear and Angular speed The second hand of a clock is 10.2 cm long. Find the linear speed of the tip of the second hand.
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A 15 inch diameter tire on a car makes 9.3 revolutions per second. a.) Find the angular speed in radians per second. b.) Find the linear speed of the car.
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The circular blade on a saw rotates at 3000 revolutions per minute. a.) Find the angular speed in radians per second. b.) The blade has a radius of 4 inches. Find the linear speed of the blade tip in inches per second.
The circular blade on a saw rotates at 3000 revolutions per minute. a.) Find the angular speed in radians per second. b.) The blade has a radius of 4 inches. Find the linear speed of the blade tip in inches per second.
a.) b.)
radians per second inches per second
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Linear Speed =
Total distance (arc length) Total time
=
(circumf) (# of revs)
Total time
dist. travelled per one unit of time
Angular Speed =
Total radians Total time
=
how fast the angle is changing how fast an object is traveling around the circle the number of radians the angle rotates through in one unit of time
Linear speed = (ang speed)(radius) (very good to know, makes work much easier when ang speed is known)
1.) Find the angular velocity in radians/min. for a spoke on a bike tire revolving 60 times per minute. 120
radians per minute
2.) The radius of the tire of a car is 18 inches, and the tire is rotating at the rate of 673 revolutions per minute. How fast is the car traveling in miles per hour? (36 )(673) inches minute 3.) A ball on the end of a string is spinning around a circle with a radius of 5 centimeters. If in 5 seconds a central angle of 1/18 radian has been covered, what is the angular speed of the ball? What is the linear speed? angular speed = .01 rads per second linear speed = (.01)(5) = .05 cm per sec
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