2. Radians and Angles Welcome to Trigonometry!! Starring The Coterminal Angles Sine Cosine Tangent Cosecant Cotangent Secant Angles Radian Degree

3. 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.

4. 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 “)

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

6. Convert 34.8 0 to DMS We need to convert the fractional part to minutes

7. Convert 112.42 0 to DMS Convert the fractional part Convert the fractional part of the minutes into seconds

8. Convert 42 0 24 ’ 36 ’’ to degrees This is the reverse of the last example. Instead if multiplying by 60, we need to divide by 60

9. Radian Measure 1 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!

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

11. Radian Measure 1 Half of a revolution (180 0 ) is equivalent to radians

12. Radian Measure 1 One fourth of a revolution (90 0 ) is equivalent to radians

13. Since there are 2π radians per 360 0, we can come up with the conversion ratio of Which reduces to radians degrees radians degrees

14. To convert degrees to radians multiply by radians degrees

15. To convert radians to degrees multiply by radians degrees

16. To convert 90 0 to radians we can multiply radians 2

17. We also know that 90 0 is ¼ of 2π radians

18. 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 THIS FORMULA ONLY WORKS WHEN THE ANGLE MEASURE IN IS RADIANS!!!

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

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

21. 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.

22. Q: Can we ever rotate the initial side counterclockwise more than one revolution? Answer – YES! EXIT BACKNEXT

23. Note: Complete Revolutions Rotating the initial side counter- clockwise 1 rev., 2 revs., 3revs.,... generates the angles which measure 360 , 720 , 1080 ,... EXIT BACKNEXT

24. Picture EXIT BACKNEXT

25. ANGLES 360, 720, & 1080 ARE ALL COTERMINAL ANGLES!

26. What if we start at 30  and now rotate our terminal side counter- clockwise 1 rev., 2 revs., or 3 revs. EXIT BACKNEXT

27. Coterminal Angles:Two angles with the same initial and terminal sides Find a positive coterminal angle to 20º Find 2 coterminal angles to Find a negative coterminal angle to 20º

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

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

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

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

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

33. Express 50.525  in degrees, minutes, seconds 50º +.525(60) 50º + 31.5 50º + 31 +.5(60)  50 degrees, 31 minutes, 30 seconds

34. CW/HW Page 280-281 (1, 3, 5-8, 11-14, 30- 33)