1. Welcome to Trigonometry!!
Radians and Angles
Welcome to Trigonometry!!
Starring
Angles
Radian
Degree
The Coterminal Angles
Sine
Cosine
Tangent
Cosecant
Cotangent
Secant
Download Angles and Angle Measure HW

2. Degree Measure The number system we use today is based on 10
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.

3. 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 “)

4. Since there are 60 ‘ in 1 degree we can convert degrees to minutes by multiplying by the conversion ratio

5. Convert to DMS
We need to convert the fractional part to minutes

6. Convert to DMS
Convert the fractional part
Convert the fractional part of the minutes into seconds

7. Convert 42024’36’’ to degrees
This is the reverse of the last example. Instead if multiplying by 60, we need to divide by 60

8. 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

9. Radian Measure
Since the circumference of a circle is 2π radians, 2π radians is equivalent to 360 degrees 1

10. Radian Measure
Half of a revolution (1800) is equivalent to
radians 1

11. Radian Measure One fourth of a revolution (900) is equivalent to
radians 1

12. Since there are 2π radians per 3600, we can come up with the conversion ratio of
degrees
radians
Which reduces to
degrees

13. To convert degrees to radians multiply by

14. To convert radians to degrees multiply by

15. To convert 900 to radians we can multiply2
radians

16. We also know that 900 is ¼ of 2π
radians

17. 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
THIS FORMULA ONLY WORKS WHEN THE ANGLE MEASURE IN IS RADIANS!!! r s θ

18. Angle-
formed by rotating a ray about its endpoint (vertex)
Terminal Side
Ending position
Initial Side
Starting position
Initial side on positive x-axis and the vertex is on the origin
Standard Position

19. 120° –210° Angle describes the amount and direction of rotation
Positive Angle- rotates counter-clockwise (CCW)
Negative Angle- rotates clockwise (CW)

20. 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.

21. Q: Can we ever rotate the initial side counterclockwise more than one revolution?
Answer – YES!
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22. Note: Complete Revolutions Rotating the initial side counter-clockwise
1 rev., 2 revs., 3revs., . . .
generates the angles which measure
360, 720, 1080, . . .
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23. Picture
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24. ANGLES 360, 720, & 1080 ARE ALL COTERMINAL ANGLES!

25. What if we start at 30 and now rotate our terminal side counter-clockwise 1 rev., 2 revs., or 3 revs.
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26. 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
Find 2 coterminal angles to

27. Warm Up
Convert to Degrees minutes, seconds
Convert to Radians:

28. Now, you try…
Find two coterminal angles (+ & -) to
What did you find?
These are just two possible answers. Remember…there are more! 

29. Complementary Angles: Two angles whose sum is 90
Supplementary Angles: Two angles whose sum is 180

30. Convert to radians: To convert from degrees radians, multiply by
To convert from degrees radians, multiply by
To convert from radians degrees, multiply by
Convert to radians:

31. So, you think you got it now?
To convert from degrees radians, multiply by
To convert from radians degrees, multiply by
Convert to degrees:
So, you think you got it now?

32. 50 degrees, 31 minutes, 30 seconds
Express  in degrees, minutes, seconds
50º (60) 
50º 
50º + 31 + .5(60) 
50 degrees, 31 minutes, 30 seconds

33. CW/HW
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