An angle with measure one radian intercepts an arc whose length is

An angle with measure one radian intercepts an arc whose length is
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.
March 15, 2017
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°
March 15, 2017
Rotation Radian measure of angle
Degree measure
1
1/2
2∏
∏
360°
180°
1/4
1/6
∏/2
∏/3
90°
60°
March 15, 2017
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
March 15, 2017
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
March 15, 2017
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
March 15, 2017
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.
March 15, 2017
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
March 15, 2017
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 =
March 15, 2017
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
March 15, 2017
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
March 15, 2017
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.
March 15, 2017
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.
March 15, 2017
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
March 15, 2017
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
March 15, 2017
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