Calculus Chapter P1 The Cartesian Plane and Functions Calculus Chapter P.
1 of 21 Pre Calculus Chapter 4.1 Warm - up. Chapter 4 Sec 1 Angles and Degree Measure.
-
Upload
kristopher-newman -
Category
Documents
-
view
217 -
download
0
Transcript of 1 of 21 Pre Calculus Chapter 4.1 Warm - up. Chapter 4 Sec 1 Angles and Degree Measure.
1 of 21
Pre Calculus Chapter 4.1
Warm - up
112112 .1 xx
223 .2 2 xxx
234
24
2
32 .3
12 .2
1214 .1
xxx
xx
x
completelyFactor
31 .3 2 xxx
3 of 21
Pre Calculus Chapter 4.1
Essential Question
How do you describe angles and angular movement?
Key Vocabulary:Initial side
Terminal side
Linear speed
Angular speed
4 of 21
Pre Calculus Chapter 4.1
• An angle in standard position has its vertex at the origin and initial side on the positive x–axis.
initial side
terminal side
Standard Position
5 of 21
Pre Calculus Chapter 4.1
• Angles that have a counterclockwise rotation have a positive measure.
positive
0º or 2π
90º or π/2
180º or π
270º or 3π/2
Positively Counterclockwise
6 of 21
Pre Calculus Chapter 4.1
• Angles that have a clockwise rotation have a negative measure.
Negative
Clockwise means negative
0º or –2π
– 270º or –3π/2
–180º or –π
–90º or – π/2
7 of 21
Pre Calculus Chapter 4.1
• Coterminal angles are angles that have the same initial and terminal side, but differ by the number of rotations.
• Since one rotation equals 2π, the measures of coterminal angles differ by multiples of 2π.
Coterminal Angles
3
3
72
3
3
5
32
3
5
8 of 21
Pre Calculus Chapter 4.1
Radian Measure
• The measure of an angle is determined by the amount of rotation from the initial side to the terminal side.
• One way to measure angles is in radians.
9 of 21
Pre Calculus Chapter 4.1
Radian…still
Since the circumference of a circle is 2πr units.
radians 2 2 rs
radians. 24
2srevolution
4
1
radians 2
2srevolution
2
1
10 of 21
Pre Calculus Chapter 4.1 Example 1
a. For the positive angle subtract
2π to obtain a coterminal angle.
b. For the positive angle subtract
2π to obtain a coterminal angle.
c. For the negative angle add 2π
to obtain a coterminal angle.
6
13
4
3
2
6
13
6
2
4
3
4
5
3
2
2
3
2
3
4
11 of 21
Pre Calculus Chapter 4.1 Complementary and Supplementary Angles
Two positive angles α and β are complementary if their sum is π/2 or 90°.
Two positive angles are supplementary if their sum is π or 180°.
Find the complement and supplement of the following
6 a.
6
5 b.complement supplement complement supplement
62
66
3
36
2
6
66
6
6
5
6
5
6
5
6
6
6
Because 5π/6 is greater than π/2 it has no complement. (Remember complements are positive angles.)
12 of 21
Pre Calculus Chapter 4.1
Guided Practice
Determine two coterminal angles in radian measure (one positive and one negative) for each angle.
6 a.
3
2 b.
Find (if possible) the complement and supplement of the angle.
6
11,
6
13 3
4,
3
8
3 c.
4
3 d.
complement
supplement3
26
4
None
13 of 21
Pre Calculus Chapter 4.1
• A second way to measure angle is in the terms of degree, denoted by °.
• One degree = 1/360 • To measure angles it is convenient to mark degrees on a
circle. So a full revolution is 360°, a half is 180° a quarter is 90°…
• We see 360° is one revolutionso 360°= 2π rad and 180° = π rad
Thus
Degree
180
rad 1 and rad 180
1
14 of 21
Pre Calculus Chapter 4.1
Degree/Radian Conversion
degree180
Radians radians180
Degree
30°
45° 60°
90°
180° 360°
15 of 21
Pre Calculus Chapter 4.1
Example 2
a. 135o
b. 540°
c. – 270°
d.
e.
f.
deg 180
rad deg 135
Convert from degrees to radians.
Convert from radians to degrees.
radians 4
3π
deg 180
rad deg 540
radians 3π
deg 180
rad deg 270
radians
2
3π
radians 2
π
radians 2
radians 2
9π
rad
deg 180rad
2
90
rad
deg 180rad 2
59.114
360
rad
deg 180rad
2
9
810
16 of 21
Pre Calculus Chapter 4.1
Arc Length
Arc LengthFor a circle of radius r, a central angle θ intercepts an arc of length s is given bys = r θ Length of circular arc where θ is measured in radians. Note that if r = 1, then s = θ, and the radian measurement of θ equal the arc length
A circle has a radius of 4 inches. Find the length of the arc intercepted by central angle of 240°.
deg 180
rad deg 240240
radians 3
4
3
44
rs inches 16.76 3
16
17 of 21
Pre Calculus Chapter 4.1 Linear and Angular Speed
Linear and Angular Speed
Consider a particle moving at a constant speed along a circular arc of radius r. If s is the length of the arc traveled in time t, then the linear speed of the particle is:
Linear speed =
Moreover, if θ is the angle (in radian measure) corresponding to the arc length s, then the angular speed of the particle is
Angular speed =
Linear speed measures how fast the particle moves and angular speed measures how fast the angle changes.
t
s
time
length arc
t
time
angle central
18 of 21
Pre Calculus Chapter 4.1
Example 4
The second hand of a clock is 10.2 centimeters long. Find the linear speed of the tip of this second hand.
In one revolution, the arc length traveled is
The time required for one revolution of the second hand is t = 1 minute or 60 seconds
So the linear speed of the tip of the second hand is
rs 2 2.102 4.20
t
sspeedLinear cm/sec 07.1
60
4.20
19 of 21
Pre Calculus Chapter 4.1
Example 5
A 15-inch diameter tire on a car makes 9.3 revolutions per second.a. Find the angular speed of the tire in radians per second.b. Find the linear speed of the car.
Because each revolution generates 2π radians, it follows that the tire turns (9.3)(2π)= 18.6π radians per second. So the angular speed is length traveled is
The linear speed of the tire is
secondper radians 6.18second 1
radians 6.18speedAngular
t
t
r
t
s speedLinear
in/sec 25.438
1
6.181521
20 of 21
Pre Calculus Chapter 4.1
Essential Question
How do you describe angles and angular movement?