1. 37: The graphs of sin  and cos  © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules

2. Graphs ofand Module C2 "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"

3. Graphs ofandTrig Functions and Graphs We are going to sketch the graph of where is an angle between and.

4. Graphs ofand P (x, y) x y O Let P be a point with coordinates (x, y) on the circle with centre at the origin and radius 1. Then, y Let angle PON be N

5. Graphs ofand y 1 20406080 x y y e.g.

6. Graphs ofand y 1 20406080 x y y e.g.

7. Graphs ofand y 1 20406080 x y y e.g.

8. Graphs ofand y 1 20406080 y e.g. x y

9. Graphs ofand y 1 20406080 x y y e.g.

10. Graphs ofand y 1 20406080 x y Also, when x x x x x x

11. Graphs ofand x x x x x x x y y 1 20406080

12. Graphs ofand As increases from to, y increases from 0 to 1. y 1 2040608010 30 5070

13. Graphs ofand We will now extend the definitions for an angle which can be any size and may be positive or negative. We have, so far, only defined for angles between and.

14. Graphs ofand x y y Suppose, then

15. Graphs ofand e.g. suppose, then x y

16. Graphs ofand x y y If,...

17. Graphs ofand x y If,... y... then y is still But So,

18. Graphs ofand The quarter of the circle where the angles are between and is called the 1 st quadrant. For each angle in the 1 st quadrant there is an angle of in the 2 nd quadrant where

19. Graphs ofand x y e.g. Notice the symmetry about the y -axis.

20. Graphs ofand x x e.g. The symmetry about enables us to draw the graph for between and.

21. Graphs ofand The symmetry about enables us to draw the graph for between and. x x e.g.

22. Graphs ofand The graph of between and is

23. Graphs ofandFor angles between and ( the 3 rd and 4 th quadrants ), the y coordinates are negative. e.g. x y So,

24. Graphs ofand The sine of each angle between and has the same magnitude as an angle between and BUT is negative. The graph for is:

25. Graphs ofand For angles larger than we continue round the circle, so the values of repeat every. e.g. y x

26. Graphs ofand We can extend the graph as shown

27. Graphs ofand We can extend the graph as shown

28. Graphs ofand Negative angles are defined as those formed by turning in a clockwise direction from. e.g. y x So,

29. Graphs ofand If you have a graphical calculator, this graph will be one of the standard graphs BUT make sure you can also sketch it without your calculator ! We can now draw the graph of for any interval. e.g.

30. Graphs ofandSUMMARY The trig function is defined for any angle. The graph of repeats every. The minimum value of is and the maximum is. The graph for is...... and must be memorised.

31. Graphs ofandExercises 1. Sketch the graph of for the interval (a) Write down an angle between and ( not equal to the given angle! ) where (b) (c)(d) (a) Ans: xx

32. Graphs ofand 1. Sketch the graph of for the interval (a) Write down an angle between and ( not equal to the given angle! ) where (b) (c)(d) (a) Ans: x x (b)Exercises

33. Graphs ofand 1. Sketch the graph of for the interval (a) Write down an angle between and ( not equal to the given angle! ) where (b) (c)(d) (a)(b)(c) Ans: x x x Exercises

34. Graphs ofand 1. Sketch the graph of for the interval (a) Write down an angle between and ( not equal to the given angle! ) where (b) (c)(d) (a)(b)(c)(d) Ans: xx x Exercises

35. Graphs ofand Sketch a circle with radius 1 and centre at the origin O. Mark a point P (x, y) on the circumference in the 1 st quadrant. Using the sketch, write down an expression for. 2(a) (b) Mark, the angle between the radius OP and the x -axis. (c) (d) (f) Refer to the circle to determine symmetry and hence complete the graph for. Use the sketch to estimate enough values of to sketch the graph of for. (e) Exercises

36. Graphs ofand 2. Solution: P (x, y) x y ON x Exercises

37. Graphs ofand 2. Solution: x y O As increases, x decreases from 1 to 0 x = 1 x = 0 e.g. When we sketch the graph we use y instead of x. Exercises

38. Graphs ofand 2. Solution: Notice that is symmetric about ( the y -axis ). Exercises

39. Graphs ofand

40. Graphs ofand The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

41. Graphs ofand The quarter of the circle where the angles are between and is called the 1 st quadrant. For each angle in the 1 st quadrant there is an angle of in the 2 nd quadrant where  )180(sin 

42. Graphs ofand Notice that is symmetric about ( the y -axis ).

43. Graphs ofand The trig functions and are defined for any angle. The graphs repeat every. The minimum value is and the maximum is. The graphs for are