radians and angles welcome to trigonometry!! starring the coterminal angles sine cosine tangent...
TRANSCRIPT
![Page 1: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/1.jpg)
Radians and AnglesWelcome to Trigonometry!!
StarringThe Coterminal Angles
Sine
Cosine
Tangent
Cosecant
Cotangent
Secant
Angles
Radian
Degree
![Page 2: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/2.jpg)
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.
![Page 3: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/3.jpg)
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 “)
![Page 4: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/4.jpg)
Since there are 60 ‘ in 1 degree we can convert degrees to minutes by multiplying by the conversion ratio
0
'
1
60
![Page 5: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/5.jpg)
Convert 34.80 to DMS
We need to convert the fractional part to minutes
'48608.
'00 48348.34
![Page 6: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/6.jpg)
Convert 112.420 to DMS
Convert the fractional part
'2.256042. Convert the fractional part of the minutes into seconds
''12602. '''00 122511242.112
![Page 7: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/7.jpg)
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
![Page 8: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/8.jpg)
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!
![Page 9: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/9.jpg)
Radian Measure
1
Since the circumference of a circle is 2π radians, 2π radians is equivalent to 360 degrees
![Page 10: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/10.jpg)
Radian Measure
1
Half of a revolution (1800) is equivalent to
22
1radians
![Page 11: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/11.jpg)
Radian Measure
1
One fourth of a revolution (900) is equivalent to
24
22
4
1 radians
![Page 12: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/12.jpg)
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
![Page 13: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/13.jpg)
To convert degrees to radians multiply by
180
radians
degrees
![Page 14: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/14.jpg)
To convert radians to degrees multiply by
180
radians
degrees
![Page 15: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/15.jpg)
To convert 900 to radians we can multiply
00
18090
radians
2
2900
radians
![Page 16: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/16.jpg)
We also know that 900 is ¼ of 2π
24
22
4
1 radians
![Page 17: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/17.jpg)
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!!!
![Page 18: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/18.jpg)
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
![Page 19: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/19.jpg)
Angle describes the amount and direction of rotation
120° –210°
Positive Angle- rotates counter-clockwise (CCW)
Negative Angle- rotates clockwise (CW)
![Page 20: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/20.jpg)
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.
![Page 21: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/21.jpg)
Q: Can we ever rotate the initial side counterclockwise more than one revolution?
Answer – YES!
EXITBACK NEXT
![Page 22: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/22.jpg)
Note: Complete Revolutions
Rotating the initial side counter-clockwise
1 rev., 2 revs., 3revs., . . .
generates the angles which measure
360, 720, 1080, . . .
EXITBACK NEXT
![Page 23: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/23.jpg)
Picture
EXITBACK NEXT
![Page 24: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/24.jpg)
ANGLES 360, 720, & 1080 ARE ALL COTERMINAL
ANGLES!
![Page 25: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/25.jpg)
What if we start at 30 and now rotate our terminal side counter-clockwise 1 rev., 2 revs., or 3 revs.
EXITBACK NEXT
![Page 26: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/26.jpg)
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
![Page 27: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/27.jpg)
Warm Up
• Convert to Degrees minutes, seconds
• Convert to Radians:
225 72
735.15
![Page 28: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/28.jpg)
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!
![Page 29: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/29.jpg)
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
![Page 30: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/30.jpg)
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
![Page 31: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/31.jpg)
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?
![Page 32: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/32.jpg)
Express 50.525 in degrees, minutes, seconds
50º + .525(60) 50º + 31.5
50º + 31 + .5(60)
50 degrees, 31 minutes, 30 seconds
![Page 33: Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree](https://reader036.vdocument.in/reader036/viewer/2022081501/56649f515503460f94c73b6a/html5/thumbnails/33.jpg)
CW/HW
• Page 280-281 (1, 3, 5-8, 11-14, 30-33)