Transcript 12 Trig Vocab PPT

Trigonometry
Trig Definitions
Radian Measure
• Recall, in the trigonometry
powerpoint, I said that Rad is Bad.
• We will finally learn what a Radian
is and how it compares to a
degree.
Radian Measure
•
One Radian is defined as the measure
of an angle that, if placed with the
vertex at the center of the circle,
intersects an arc of length equal to the
radius of the circle.
•
One radian measure where the radius and the arc length
are the same length.
•
Therefore, in this diagram
rb
The Circle and the
Radian
•
The circumference of a
circle.
C  2 r
Circumference of a
Unit Circle
What is the circumference
of a unit circle where the
radius = 1?
Circumference of the
Unit Circle
•
If r
1
the circumference of a unit circle is
C  2 (1)
C  2
Relationship between
Degrees and Radians
2
If the circumference of a unit circle is
and if a circle has 360
degrees what is the equivalent radian measure for the following:
90 degrees =
180 degrees =
270 degrees =
360 degrees =
radians
radians
radians
radians
Answers
90 degrees =
180 degrees =
270 degrees =

2

3
2
radians
radians
radians
Converting from
Degrees to Radians
Unit Analysis:
Recall if you want to cancel
degrees on top you must have
degrees on the bottom to cancel
the units. Think of the degree
symbol as a ‘unit’ that needs.
1 
o

180
o
Convert Degrees to
Radians
Convert 210 degrees to Radian Measure using the ratio provided on the
previous slide.
210  210 
o

180
Warning:
radians
o
7

radians
6
Do
not use
decimals to
simplify!!
Convert the following degree measures to radian measures:
30 degrees
45 degrees
60 degrees
90 degrees
120 degrees
135 degrees
150 degrees
180 degrees
270 degrees
360 degrees
Warning: Do
not use
decimals to
simplify!!
Convert the following degree measures to radian measures:
30 degrees
45 degrees
60 degrees

90 degrees 6

120 degrees4

3

2
135 degrees 2
3
150 degrees
3
4
5
6

3
2
2
Warning: Do
not use
decimals to
simplify!!
Converting from
Radians to Degrees
1 radian 
180

o
Think about unit analysis
again in this conversion. If
you need to convert to
degrees you need to have
radians on the bottom and
degrees left on the top.
Convert 6 Radians
to Degrees
6 radians  6 
180

 6  180
 1080
o
o
o
Arc Length
•
To calculate arc length use the formula:
s  r
Arc Length
Example
•
Determine the radius of a circle in which a central
angle of 3 radians subtends an arc of length 30 cm.
s  r
30  r  10
r3
Coterminal Angles
•
Two angles in standard position are
coterminal if they have the same terminal
side. There are infinite number of angles
coterminal with a given angle.
•
To find an angle coterminal with a given
angle, add or subtract
2
•
For example,
  2  3
Angles
• A trigonometric angle is determined by rotating a ray
about is endpoint, called the vertex of the angle
• The starting position of the ray is called the initial side
and the ending position is the terminal side
Terminal Side
Initial Side
Initial and Terminal
Sides
Which are is the initial side and
which are is the terminal side?
Angle Direction
•
If the displacement of the ray from its starting position is
in the counter clockwise position it is assigned a positive
measure
•
If the displacement of the ray from its starting position is
in the clockwise position it is assigned a negative
measure
Standard Position
• An angle is in standard position in a
Cartesian Coordinate system if its
vertex is at the origin and it initial
side is the positive x-axis.
Standard Position
Graph