3.1 -- Radian Measures
Wednesday, August 07, 2013
1:00 PM
Radian Measure: We have seen that angles can be measured in degrees. In more theoretical
work in mathematics, radian measure of angles is preferred.
An angle with its vertex at the center of a circle that intercepts
an arc on the circle equal in length to the radius of the circle has
a measure of 1 radian.
***So θ = 1 radian when the arc length = radius
Converting between Degrees and Radians
The circumference of a circle, C = 2πr and since a complete circle measures 360º, then
360° = 2π radians
Degree
Radian multiply by
180° = π radians
Radian
Degree multiply by
Ex1: Converting from Degrees to Radians
1) 45°
2) 270°
3) 249.8°
4) 108°
6) 325.7°
5)
Ex2: Converting Radians to Degrees
1)
Chapter 3 Page 1
135°
If no unit of angle measure is specified, then the angle is understood to be measured in radians.
** in calculus, radians is
much easier to use!
30 radians
The Unit Circle
30 degrees
You need to learn these equivalences
ASAP! They will be used very often
throughout the year.
The 30⁰ reference angles go with
The 45⁰ reference angles go with
The 60⁰ reference angles go with
Ex3: Finding Trig Function Values for Angles in Radians
1)
2)
Chapter 3 Page 2
3)
3.2 -- Applications of Radian Measure
Wednesday, August 07, 2013
1:00 PM
Arc Length on a Circle = s
s = rθ
θ must be in radians
Ex1: Finding Arc Length using s = rϴ
A circle has a radius of 18.2 cm. Find the length of the arc intercepted by a central angle
having each of the following measures.
2) 144⁰
1)
4) 54⁰
Ex2: Finding the Distance Between 2 Cities
Latitude gives the measure of a central angle with vertex at
Earth's center whose initial side goes thru the equator and
whose terminal side goes thru the given location. Reno, Nevada,
is approx due north of L.A. The latitude of Reno is 40⁰ N, and that
of L.A. is 34⁰ N. (The N in 34⁰ N means north of the equator.) The
radius of the Earth is 6400 km. Find the north-south distance
between the 2 cities.
Chapter 3 Page 3
Reno
s
LA
6º
40º
34⁰
Equator
Classroom Ex2: Erie, Pennsylvania, is approx due north of Columbia. The latitude of Erie is
42⁰ N, and that of Columbia is 34⁰ S. The radius of the Earth is 6400 km. Find the northsouth distance between the 2 cities.
Ex3: Find a Length Using s = rθ
A rope is being wound around a drum with a radius 0.8725 ft. How much rope will be wound
around the drum if the drum is rotated thru an angle of 39.72⁰?
Classroom Ex3 with r = 0.327 m and the drum is rotated thru an angle of 132.6°.
Ex4
Chapter 3 Page 4
Classroom Ex4: If the radii of the gears are 3.6 in. and 5.4 in. and the smaller gear rotates
thru 150⁰.
** the θ must be in radians
Area of a sector of a circle:
Ex5: Finding the Area of a Sector-Shaped Field
A center-pivot irrigation system provides water to a sector-shaped field with measures shown
below. Find the area of the field.
321 m
15º
Classroom Ex5: Find the area of a sector of a circle having radius 15.2 ft and central angle 108⁰.
Find the measure of each central angle (in radians). s = rϴ
20
6
3
ϴ
ϴ
ϴ
3
4
Chapter 3 Page 5
10
3.3 -- The Unit Circle and Circular Functions
Wednesday, August 07, 2013
1:01 PM
Circular Functions
** if you plan to study calculus, you must become very familiar with radian measure. In calculus, the trig or
circular functions are always understood to have real number domains
Since cos s = x and sin
s = y and x2 + y2 = 1,
then
x = cos s
y = sin s
cos2 s + sin2 s = 1
Finding Values of Circular Functions is done the same way as finding the trig functions of angles
measured in radians. If you use a calculator it must be in radian mode.
Chapter 3 Page 6
(0, 1)
Ex1: Finding Exact Circular Function Values
Find the exact values of
(-1, 0)
(1, 0)
(0, -1)
Classroom Ex1: Find the exact values of sin ( 3π), cos ( 3π), and cot ( 3π).
Ex2: Finding Exact Circular Function Values
Find each exact value using the specified method
1) Use the unit circle to find the exact values of
and
2) Use the unit circle and the definition of the tangent to find the exact value of
3) Use reference angles and radian-to-degree conversion to find the exact value of
Classroom Ex2:
Chapter 3 Page 7
Classroom Ex2:
1) Use the unit circle to find the exact values of
and
2) Use the unit circle and the definition of the tangent to find the exact value of
3) Use reference angles and radian-to-degree conversion to find the exact value of
Ex3: Approximating Circular Function Values (remember to put your calculator in radian mode)
Use a calculator to approx
1) cos 1.85
2) cos 0.5149
3) cot 1.3209
4) sec ( 2.9234)
5) sin 3.42
6) tan 0.8234
7) sec 5.6041
8) csc ( 2.5198)
Ex4: Finding an angle measure given its circular function value (you will use inverse functions)
You can use degrees or radians to work out the problem, but your final answer must be in radians.
The interval given will tell you in what quadrant your answer must be.
1) Approximate the value of s in the interval
2) Find the exact value of s in the interval
Chapter 3 Page 8
if cos s = 0.9685.
if tan s = 1.
Classroom Ex4:
1) Approximate the value of s in the interval
2) Find the exact value of s in the interval
Chapter 3 Page 9
if sin s = 0.9685.
if tan s =
.
Quiz review?
Tuesday, August 13, 2013
7:33 PM
Chapter 3 Page 10
3.4 -- Linear and Angular Speed
Wednesday, August 07, 2013
1:01 PM
• Linear speed (v) is how
fast point P is changing.
P
• Angular speed (ω) is how
fast the POB is changing.
or
O
B
s = arc length
s = rϴ
ω is the rate of angular speed
measured in radians per unit
of time
This formula is just a restatement of d = rt
Angular Speed Example
The wrist is the fastest flexing human joint.
It can rotate thru 90⁰, or radians, in 0.045
0.045 sec while holding a tennis racket.
Linear Speed Example
If the radius (distance) from the tip of the
racket to the wrist joint is 2 ft, then the
speed at the tip of the racket is
Ex1: Using Linear an Angular Speed Formulas
Point P is on a circle with radius 10 cm, and
is rotating with angular speed
1) Find the angle generated by P in 6 sec.
2) Find the distance traveled by P along the circle in 6 sec.
3) Find the linear speed of P in cm per sec.
Classroom Ex1:
Chapter 3 Page 11
radian per sec.
Classroom Ex1:
Point P is on a circle with radius 15 in., and
is rotating with angular speed
1) Find the angle generated by P in 10 sec.
2) Find the distance traveled by P along the circle in 10 sec.
3) Find the linear speed of P in inches per sec.
radian per sec.
Ex2: Finding Angular Speed of a Pulley and Linear Speed of a Belt
A belt runs a pulley of radius 6 cm at 80 revolutions per min.
1) Find the angular speed of the pulley in radians per second.
2) Find the linear speed of the belt in centimeters per second.
Classroom Ex2: Repeat for a belt that runs a pulley of radius 5 in. at 120 revolutions per min.
Chapter 3 Page 12
Ex3: Finding Linear Speed and Distance Traveled by a Satellite
A satellite traveling in a circular orbit 1600 km above the surface of Earth takes
2 hr to make an orbit. The radius of Earth is approx 6400 km.
1600 km
1) Approx the linear speed of the satellite in km per hour.
6400 km
2) Approx the distance the satellite travels in 4.5 hr.
not drawn to scale
Classroom Ex3:
A satellite traveling in a circular orbit 1800 km above the surface of Earth takes 2.5 hr to make
an orbit. The radius of Earth is approx 6400 km.
1) Approx the linear speed of the satellite in km per hour.
2) Approx the distance the satellite travels in 3.5 hr.
Chapter 3 Page 13
Test review?
Tuesday, August 13, 2013
7:33 PM
Chapter 3 Page 14