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Radians and AnglesWelcome to Trigonometry!!
StarringThe Coterminal Angles
Sine
Cosine
Tangent
Cosecant
Cotangent
Secant
Angles
Radian
Degree
Degree MeasureOver 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
0
'
1
60
Convert 34.80 to DMS
We need to convert the fractional part to minutes
'48608.
'00 48348.34
Convert 112.420 to DMS
Convert the fractional part
'2.256042. Convert the fractional part of the minutes into seconds
''12602. '''00 122511242.112
Convert 42024’36’’ to degrees
This is the reverse of the last example. Instead if multiplying by 60, we need to divide by 60
000
0'''0 41.426060
36
60
2442362442
Radian Measure
1
The circumference of a circle is 2πrIn 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!
Radian Measure
1
Since the circumference of a circle is 2π radians, 2π radians is equivalent to 360 degrees
Radian Measure
1
Half of a revolution (1800) is equivalent to
22
1radians
Radian Measure
1
One fourth of a revolution (900) is equivalent to
24
22
4
1 radians
Since there are 2π radians per 3600, we can come up with the conversion ratio of
360
2
180
Which reduces to
radians
degrees
radians
degrees
To convert degrees to radians multiply by
180
radians
degrees
To convert radians to degrees multiply by
180
radians
degrees
To convert 900 to radians we can multiply
00
18090
radians
2
2900
radians
We also know that 900 is ¼ of 2π
24
22
4
1 radians
Arc length formula
θ
r
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
rs THIS FORMULA ONLY WORKS WHEN THE ANGLE MEASURE IN IS RADIANS!!!
Angle- formed by rotating a ray about its endpoint (vertex)
Initial Side Starting position
Terminal Side Ending position
Standard PositionInitial side on positive x-axis 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!
EXITBACK NEXT
Note: Complete Revolutions
Rotating the initial side counter-clockwise
1 rev., 2 revs., 3revs., . . .
generates the angles which measure
360, 720, 1080, . . .
EXITBACK NEXT
Picture
EXITBACK NEXT
ANGLES 360, 720, & 1080 ARE ALL COTERMINAL
ANGLES!
What if we start at 30 and now rotate our terminal side counter-clockwise 1 rev., 2 revs., or 3 revs.
EXITBACK NEXT
Coterminal Angles: Two angles with the same initial and terminal sides
Find a positive coterminal angle to 20º 38036020
34036020
Find 2 coterminal angles to 4
15
4
8
4
15
24
15
4
8
4
15
24
15
4
23
4
8
4
7
Find a negative coterminal angle to 20º
4
Warm Up
• Convert to Degrees minutes, seconds
• Convert to Radians:
225 72
735.15
Now, you try…
Find two coterminal angles (+ & -) to 3
2
What did you find?
3
8,
3
4
These are just two possible answers. Remember…there are more!
Complementary Angles: Two angles whose sum is 90
Supplementary Angles: Two angles whose sum is 180
6
62
36
2
66
3
3
2
3
233
2
3
3
To convert from degrees radians, multiply by
To convert from radians degrees, multiply by
180
180
Convert to radians:
180
135
4
3
180
80
9
4
To convert from degrees radians, multiply by
To convert from radians degrees, multiply by
180
180
Convert to degrees:
180
3
8 480
180
6
5 150
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
• Page 280-281 (1, 3, 5-8, 11-14, 30-33)