Transcript Welcome to Trigonometry!! Radians and Angles Starring 

Radians and Angles
Welcome to Trigonometry!!
Starring
Angles
The Coterminal Angles
Radian
Sine
Degree
Cosine

Tangent
Cosecant
Cotangent
Secant
Degree Measure
Over 2500 years ago, the Babylonians used a
number system based on 60
The number system we use today is based
on 10
However we still use the Babylonian idea to
measure certain things such as time and angles.
That is why there are 60 minutes in an hour and
60 seconds in a minute.
The Babylonians divided a circle into
360 equally spaced units which we call
degrees.
In the DMS (degree minute second) system
of angular measure, each degree is
subdivided into 60 minutes (denoted by ‘ )
and each minute is subdivided into 60
seconds (denoted by “)
Since there are 60 ‘ in 1 degree we can
convert degrees to minutes by
multiplying by the conversion ratio
60
0
1
'
Convert 34.80 to DMS
We need to convert the fractional part to
minutes
.8  60  48
'
34.8  34 48
0
0
'
Convert 112.420 to DMS
Convert the fractional part
.42  60  25.2
'
Convert the fractional part of the
minutes into seconds
.2  60  12
''
112.42  112 25 12
0
0
'
''
Convert 42024’36’’ to degrees
This is the reverse of the last example.
Instead if multiplying by 60, we need to
divide by 60
0
0
 24   36 
0
42 24 36  42     
  42.41
 60   60  60 
0
'
''
0
Radian Measure
The circumference of a circle is 2πr
In a unit circle, r is 1, therefore the circumference is 2π
A radian is an angle measure given in terms of π. In
trigonometry angles are measured exclusively in radians!
1
Radian Measure
Since the circumference of a circle is 2π
radians, 2π radians is equivalent to 360 degrees
1
Radian Measure
Half of a revolution (1800) is equivalent to
1
 2  
2
1
radians
Radian Measure
One fourth of a revolution (900) is equivalent to
1
2 
 2 

4
4
2
1
radians
Since there are 2π radians per 3600, we can
come up with the conversion ratio of
2 radians
360 degrees
Which reduces to

radians
180
degrees
To convert degrees to radians multiply by

radians
180
degrees
To convert radians to degrees multiply by
180 degrees

radians
To convert 900 to radians we can multiply
90 

0
180
radians
0
2
90 
0

2
radians
We also know that 900 is ¼ of 2π
1
2 
 2 

4
4
2
radians
Arc length formula
If θ (theta) is a central angle in a circle of radius r,
and if θ is measured in radians, then the length s of
the intercepted arc is given by
s  r
r
s
θ
THIS FORMULA
ONLY WORKS
WHEN THE ANGLE
MEASURE IN IS
RADIANS!!!
Angle-
formed by rotating a ray
about its endpoint (vertex)
Terminal Side Ending position
Initial Side Starting position
Initial side on positive x-axis
Standard Position
and the vertex is on the origin
Angle describes the amount and direction of rotation
120°
–210°
Positive Angle- rotates counter-clockwise (CCW)
Negative Angle- rotates clockwise (CW)
Coterminal Angles
• Angles with the same initial side and same
terminal side, but have different rotations, are
called coterminal angles.
• 50° and 410° are coterminal angles. Their
measures differ by a multiple of 360.
Q: Can we ever rotate the initial side
counterclockwise more than one
revolution?
Answer – YES!
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EXIT
Note: Complete Revolutions
Rotating the initial side counterclockwise
1 rev., 2 revs., 3revs., . . .
generates the angles which measure
360, 720, 1080, . . .
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EXIT
Picture
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EXIT
ANGLES 360, 720, &
1080 ARE ALL
COTERMINAL
ANGLES!
What if we start at 30 and now
rotate our terminal side counterclockwise 1 rev., 2 revs., or 3 revs.
BACK
NEXT
EXIT
Coterminal Angles: Two angles with the same initial
and terminal sides
Find a positive coterminal angle to 20º
Find a negative coterminal angle to 20º
15
Find 2 coterminal angles to
4
20  360  380
20  360  340
23
15 8
15


 2 
4
4
4
4
15
15 8
 2 

4
4
4
7 8




4
4
4
Warm Up
• Convert to Degrees minutes, seconds
15.735
• Convert to Radians:
72
225
Now, you try…
 2
Find two coterminal angles (+ & -) to
3
What did you find?
4  8
,
3
3
These are just two possible answers.
Remember…there are more! 
Complementary Angles: Two angles whose sum is 90

6

2


 3    2  
6
6 6
6
3
Supplementary Angles: Two angles whose sum is 180
2
3
2
3 2 




3
3
3
3
To convert from degrees
To convert from radians
radians, multiply by
degrees, multiply by
Convert to radians:
135 

180
3

4
 80 

180
4

9

180
180

To convert from degrees
To convert from radians
Convert to degrees:
 8 180

  480
3

5 180


6 
150
radians, multiply by
degrees, multiply by

180
180

So, you think you
got it now?
Express 50.525 in degrees, minutes, seconds
50º + .525(60) 
50º + 31.5
50º + 31 + .5(60) 
50 degrees, 31 minutes, 30 seconds
CW/HW
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