trigonometry “measurement of triangles”. initial side angle terminal side vertex angles are...
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Trigonometry
“Measurement of triangles”
• Initial side
Angle• Terminal
side
• Vertex Angles are always labeled with either a capital letter or a Greek letter
When the initial side is on the x axis, the angle is in STANDARD POSITION
Types of angles:• Positive angles:
• Negative angles:
• Coterminal angles:
Generated by a counterclockwise rotationGenerated by a clockwise rotation
Angles that have the same initial and terminal sides
There are pictures on page 262
Measures of angles:determined by the amount of rotation
from the initial side to the terminal side• One way to measure are:
RADIANSMeasure of a central angle O that intercepts an
arc s equal in length to the radians r of the
circle
Radians…•Because the circumference of a circle is units it follows that a central angle of one full
revolution (counterclockwise) corresponds to an arc length:
r2
rs 2•A radian measure of a central angle is
r
s
Because the radian measure of an angle of one full revolution is … 2
You need to memorize!!
Q2 Q1
Q3Q4
6
6
13What can you tell me about and ?
Coterminal Angles
2 2To find coterminal
angles:
Practice finding coterminal angles:
3
24
3
Complementary and Supplementary Angles
Complementary: sum = Supplementary: sum = 2
Lets find the complement and supplement of
5
2
5
4Lets find the complement and
supplement of
Day 1 HW
pg. 269 # 1 – 23 odd
Warm Up: Complete the following chart with the correct radian measure Complete the following problems: pg. 269 #’s 4, 6, 8,
12, 14, 18
Degree MeasureThe second way to measure
angles.
What is the relationship between degrees and radians?
Fill in the following circle with the correct radian and degree
measures:
If NO unit measures are specified, then radian
measure is IMPLIED, so assume radians!!Let’s practice converting:
radians2.)560
.)4
180.)3
540.)2
135.)1
Another way to express degrees is to denote fractional parts of
degrees by minutes (‘) and seconds(”).
??? How would represent 64 degrees, 32 minutes, and 47
seconds in this form?
Convert this angle to decimal degree form:
44 15 20
44.255
To convert from decimal degree form to
form, you can use your calculator.
1. Type in decimal degree
2. 2nd angle (apps)
3. Choose 4 (DMS)
4. Enter
Try:
1.) 125.36o 2.) -4.68o
Day 2 HWpg. 269 #’s 24 – 77 by
3’s
APPLICATIONSFinding arc length:
Recall that , therefore s = ?
r
s
*always in radians
Let’s do example 5 on page 267.
s = 2πr
Will be helpful to think of θ: 2π times the number of revolutions
Some more practice:1. Find the angle measure of an angle with a
radius of 10 and arc length of 12.
2. On a circle with a radius of 9 inches, find the length of the arc intercepted by a central angle of 140 degrees.
5
6
inches227
Arc length is used to analyze motion of a particle moving at a constant speed along a circular
path.Linear speed :
Measures how fast the particle moves
Angular speed:
Measures how fast the angle changes
Practice finding linear speed…
The second hand of a clock is 10.2 cm long. Find the linear speed of the tip of this second hand as it passes around the clock face.
Some practice…1. A lawn roller with a 20 inch radius makes 2
revolutions per second. Find the linear speed in inches per second of a point on the outside of the roller after 1 second.
2. A bicycle wheel is turning at a rate of 15 revolutions per minute. The wheel is 24 inches in diameter. Find the linear velocity in feet per second of a piece of gum stuck to the outside of the wheel.
251.32 in/sec
1.57 ft/sec
Practice finding angular speed and linear
speed:A lawn roller with a 10 inch radius makes 1.2 revolutions per second. Find the angular speed of the roller in radians per second, then find the speed of the tractor that is pulling the roller (linear speed).
Some practice:1. The wheel of a truck is turning at a rate of 6
revolutions per second. The wheel is 4 feet in diameter.
a.) Find the angular speed in radians per second.
b.) Find the linear speed, in feet per second, of a point on the rim of the wheel.
12π rad/sec
75.4 ft/sec
HW – Day 3pg. 269 # 25, 29, 35, 37, 41, 49, 51, 53, 55, 57, 63, 67, 79, 81, 83,
87, 99
Day 4 Warm Uppg. 271 # 99
Day 4 – With a partnerpg. 271 # 91, 95, 96,
97, 98, 100, 102
Study problems for quiz…
Pg. 269 # 8, 14, 18, 22, 30, 38, 42, 58, 64,
68, 80, 84, 88