Transcript co-terminal angles.

13.2
General Angles and Radian
Measure
History Lesson of the Day
Hippocrates of Chois (470-410 BC) and
Erathosthenes of Cyrene (276-194 BC) began
using triangle ratios that were used by
Egyptian and Babylonian engineers 4000 years
earlier.
Term “trigonometry” emerged in the 16th
century from Greek roots.
“Tri” = three
“gonon” = side
“metros” = measure
Why Study Trig?
• Trig functions arose from the consideration of ratios
within right triangles.
– Ultimate tool for engineers in the ancient world.
• As knowledge progressed from a flat earth to a world of
circles and spheres, trig became the secret to
understanding circular phenomena.
• Circular motion led to harmonic motion and waves.
– Electrical current
– Modern telecommunications
– Store sound wave digitally on a CD
What you’ll learn about
•
•
•
•
The Problem of Angular Measure
Degrees and Radians
Arc Length
Area of a Sector
… and why
Angles are the domain elements of the
trigonometric functions.
Why 360º?
The idea of dividing a circle into 360 equal pieces dates
back to the sexagesimal (60-based) counting system of
the ancient Sumerians. Early astronomical calculations
linked the sexagesimal system to circles.
The problem is that degree
units have no mathematical
relationship whatsoever to
linear units. Therefore we
needed another way to
measure a circle.
Vocabulary
Circumference: The distance all the way around
a circle.
C  2r
What is the circumference of a circle whose
radius equals 10 feet?
C  2 *  * (10inches )  20 .inches
Your
C Turn:
2r
1. Circle #1 has a radius of 1 inch, what is the
circumference of circle #1? (leave ‘pi’ in your answer)
2. Circle #2 has a radius of 3 inches, what is the
circumference of circle #2? (leave ‘pi’ in your answer)
3. What does the ratio the circumference of
Circle #1 to Circle #2?
The ratio of circumference to radius is CONSTANT.
C
 2
r
regardless of the size of the circle.
Vocabulary
arc..length
Radian ..Measure 
radius
Radian measure: the ratio of the arc length to
the radius of the circle:
C..inches
inches
 2 ..
 2 ..Radians
r..inches
inches
Degree measure of a circle: 360º
2
Radian measurement of a circle =
Degree-Radian Conversion
Degree-Radian Conversion
180° = π radians
180
To convert radians to degrees, multiply by
.
 radians
 radians
To convert degrees to radians, multiply by
.
180
  
?

o 
 180 
 180 o 

  ?
  
These are
“conversion factors”
When you multiply a number by one
of these factors, it converts the units.
Converting from Degrees to Radian
Measure
   140
7 What property
   is used here?

o  
9
 180  180
Converting from Radian Measure to
Degrees
o
  180  90


2   
140°
Your Turn: Convert between radians and degrees.
4.
11
 ?
3
5.
240 o  ?
Initial Side, Terminal Side
final position of the ray
α, β, θ = the measure of the angle
beginning position of the ray
Vertex
Vocabulary
Standard Position An acute angle with one ray
along the x-axis and the other
ray rotated clockwise from the first ray.
Terminal Side
In Trigonometry, we
sometimes use a circle
with the vertex of the
angle at the center of
the circle.
Vocabulary
Standard Position An acute angle with one
ray along the positive x-axis and the other
ray rotated clockwise from the first ray.
Terminal Side
Initial Side
We use
the measure
of the acute angle
with the x-axis.
In Trigonometry, we
sometimes use a circle
with the vertex of the
angle at the center of
the circle.
Vocabulary
Standard Position An acute angle with one ray
along the positive x-axis and the other
ray rotated clockwise from the first ray.
Initial Side
Terminal Side
We use
the measure
of the acute angle
with the x-axis.
Angle measures
90º
Draw the angle with the given
measure in standard form.
220º
220º
180º
0º
40º
220º
270º
40º
Your turn:
Draw an angle in standard position that
has a measure of:
6. 135º
7. 290º
Co-terminal Angles
What is the difference in
position on the unit circle
if terminal side stops at
45º or goes all the way
around and stops at 405º ?
45º
Co-terminal Angles
What is the difference in
position on the unit circle
if terminal side stops at
45º or goes all the way
around and stops at 405º ?
45º
There is no difference !!
Although the angular
measure is different they
are co-terminal angles.
Finding Co-terminal Angles
Find a positive and a negative angle that are
co-terminal with 45°.
We’ve already found
one positive co-terminal
angle with 45° (405°).
Can you find another?
45º
405° + 360° = 765°
Negative angle:
45°- 360° = -315°
Your Turn:
8. Find a positive co-terminal angle with 120°
9. Find a negative co-terminal angle with 270°
Finding Co-terminal Angles
Find a positive and negative co-terminal angle
with:
Notice the angle measure is now in radians.

13
 2 
6
6

11
 2  
6
6

6
Your Turn:
10. Find a positive co-terminal angle with
11. Find a negative co-terminal angle with
2
3

2
Arc Length
Remember:
radian measure is the
ratio of arc length to radius.
Which gives us this
formula for arc length.
r


s  r
Gothcha: the angle
measure must be
in radians not degrees.
Arc Length

Radian =
= Greek letter “theta”
s  r
 arc..length 


'
 radius 
Often used as a variable to
denote the measure of
an angle.
“arc length = radius * angle measure (in radians)”
r = 5 inches


3
radians


s  (5inches ) radians 
3

Arc length = ?
5
s
inches
3
Your Turn:
3

12. radius = 10 inches,  
7
Arc length = ?
2
13. Arc length =
Radius = 6 inches
3
What is the angle measure (in radians)?
14. What is the angle measure for problem
#12 in degrees?
Sector Area
Area of a circle:
A  2r
A  20 . ft
10 ft
30º

What fraction of the circle
is a 30º sector?

30
1
Fraction.of .circle 


12
360
Sector Area =
Sector Area =
1 (20 . ft)
12
5
ft
3
Your turn:
15. Find the area of a slice of pizza.
14 inch pizza (diameter)
Slice is 1/8 of the pizza
HOMEWORK