1. Angles of Any Magnitude

2. Angles of Any Magnitude
y 90
II I
180 
0
x 360
III IV
270

3. Angles of Any Magnitude
y 90
II I
180 
0
x 360
III IV
270

4. Angles of Any Magnitude
y 90
II I
180 
  0
  x 360
III IV
270

5. Angles of Any Magnitude
y 90
II I
180 
  0
  x 360
III IV
270
Quadrant I - normal

6. Angles of Any Magnitude
y 90
II I
180 
  0
  x 360
III IV
270
Quadrant I - normal
Quadrant II - 180   

sin   


cos   


tan   


7. Angles of Any Magnitude
y 90
II I
180 
  0
  x 360
III IV
270
Quadrant I - normal
Quadrant II - 180    Quadrant III - 180   
 
sin    sin   
 
 
cos    cos   
 
 
tan    tan   
 

8. Angles of Any Magnitude
y 90
II I
180 
  0
  x 360
III IV
270
Quadrant I - normal
Quadrant II - 180    Quadrant III - 180    Quadrant IV -  360   
  
sin    sin    sin   
  
  
cos    cos    cos   
  
  
tan    tan    tan   
  

9. Angles of Any Magnitude
y 90 e.g.  i  sin 260   sin 80
II I  0.9848
180 
  0
  x 360
III IV
270
Quadrant I - normal
Quadrant II - 180    Quadrant III - 180    Quadrant IV -  360   
  
sin    sin    sin   
  
  
cos    cos    cos   
  
  
tan    tan    tan   
  

10. Angles of Any Magnitude
y 90 e.g.  i  sin 260   sin 80  ii  tan 220  tan 40
II I  0.9848  0.8391
180 
  0
  x 360
III IV
270
Quadrant I - normal
Quadrant II - 180    Quadrant III - 180    Quadrant IV -  360   
  
sin    sin    sin   
  
  
cos    cos    cos   
  
  
tan    tan    tan   
  

11. Angles of Any Magnitude
y 90 e.g.  i  sin 260   sin 80  ii  tan 220  tan 40
II I  0.9848  0.8391
180 
  0 iii  sec105   sec 75
  x 360  3.8637
III IV
270
Quadrant I - normal
Quadrant II - 180    Quadrant III - 180    Quadrant IV -  360   
  
sin    sin    sin   
  
  
cos    cos    cos   
  
  
tan    tan    tan   
  

12. Angles of Any Magnitude
y 90 e.g.  i  sin 260   sin 80  ii  tan 220  tan 40
II I  0.9848  0.8391
180 
  0 iii  sec105   sec 75  iv  cos 430  cos 70
  x 360  3.8637  0.3420
III IV
270
Quadrant I - normal
Quadrant II - 180    Quadrant III - 180    Quadrant IV -  360   
  
sin    sin    sin   
  
  
cos    cos    cos   
  
  
tan    tan    tan   
  

13. Angles of Any Magnitude
y 90 e.g.  i  sin 260   sin 80  ii  tan 220  tan 40
II I  0.9848  0.8391
180 
  0 iii  sec105   sec 75  iv  cos 430  cos 70
  x 360  3.8637  0.3420
III IV  v  tan  67   tan 293
270
Quadrant I - normal
Quadrant II - 180    Quadrant III - 180    Quadrant IV -  360   
  
sin    sin    sin   
  
  
cos    cos    cos   
  
  
tan    tan    tan   
  

14. Angles of Any Magnitude
y 90 e.g.  i  sin 260   sin 80  ii  tan 220  tan 40
II I  0.9848  0.8391
180 
  0 iii  sec105   sec 75  iv  cos 430  cos 70
  x 360  3.8637  0.3420
III IV  v  tan  67   tan 293
  tan 67
270  2.3559
Quadrant I - normal
Quadrant II - 180    Quadrant III - 180    Quadrant IV -  360   
  
sin    sin    sin   
  
  
cos    cos    cos   
  
  
tan    tan    tan   
  

15. (vi) Find the coordinates of P
y
P
30
5 x

16. (vi) Find the coordinates of P
x
 cos30 y
5
P
30
5 x

17. (vi) Find the coordinates of P
x
 cos30 y
5
x  5cos30 P
30
5 3 5 x
x
2

18. (vi) Find the coordinates of P
x y
 cos30 
 sin 30 y
5 5
x  5cos30  P
30
5 3 5 x
x
2

19. (vi) Find the coordinates of P
x y
 cos30 
 sin 30 y
5 5
x  5cos30 
y  5sin 30 P
30
5 3 5 5 x
x y
2 2
5 3 5
P , 
 2 2

20. (vi) Find the coordinates of P
x y
 cos30 
 sin 30 y
5 5
x  5cos30 
y  5sin 30 P
30
5 3 5 5 x
x y
2 2
5 3 5
P , 
 2 2
Boundary Values
y
90
180 0
x
270

21. (vi) Find the coordinates of P
x y
 cos30 
 sin 30 y
5 5
x  5cos30 
y  5sin 30 P
30
5 3 5 5 x
x y
2 2
5 3 5
P , 
 2 2
Boundary Values
90 1
y 0
90 0
180 0 1
x
270

22. (vi) Find the coordinates of P
x y
 cos30 
 sin 30 y
5 5
x  5cos30 
y  5sin 30 P
30
5 3 5 5 x
x y
2 2
5 3 5
P , 
 2 2
Boundary Values
90 1
y 0
90 0
180 0 0 1
x sin 0  0
270 cos 0  1
tan 0  0

23. (vi) Find the coordinates of P
x y
 cos30 
 sin 30 y
5 5
x  5cos30 
y  5sin 30 P
30
5 3 5 5 x
x y
2 2
5 3 5
P , 
 2 2
Boundary Values
90 1
y 0
90 0
180 0 0 1 90
x sin 0  0 sin 90  1
270 cos 0  1 cos90  0
tan 0  0 tan 90  1
0
undefined

24. (vi) Find the coordinates of P
x y
 cos30 
 sin 30 y
5 5
x  5cos30 
y  5sin 30 P
30
5 3 5 5 x
x y
2 2
5 3 5
P , 
 2 2
Boundary Values
90 1
y 0
90 0
180 0 0 1 90 180
x sin 0  0 sin 90  1 sin180  0
270 cos 0  1 cos90  0 cos180  1
tan 0  0 tan 90  1 tan180  0
0
undefined

25. (vi) Find the coordinates of P
x y
 cos30 
 sin 30 y
5 5
x  5cos30 
y  5sin 30 P
30
5 3 5 5 x
x y
2 2
5 3 5
P , 
 2 2
Boundary Values
90 1
y 0
90 0
180 0 0 1 90 180 270
x sin 0  0 sin 90  1 sin180  0 sin 270  1
270 cos 0  1 cos90  0 cos180  1 cos 270  0
tan 0  0 tan 90  1 tan180  0 tan 270  1
0 0
undefined undefined

26. Trigonometric Graphs

27. Trigonometric Graphs plot boundary values

28. Trigonometric Graphs plot boundary values
y
360 180 180 360 540 720 x
-1

29. Trigonometric Graphs plot boundary values
y
1
360 180 180 360 540 720 x
-1

30. Trigonometric Graphs plot boundary values
y
1
360 180 180 360 540 720 x
-1

31. Trigonometric Graphs plot boundary values
y
1
360 180 180 360 540 720 x
-1

32. Trigonometric Graphs plot boundary values
y
1 y  sin x
360 180 180 360 540 720 x
-1
domain : all real x
range : - 1  y  1
period: 360

33. Trigonometric Graphs plot boundary values
y
1 y  sin x
360 180 180 360 540 720 x
-1
domain : all real x odd function
range : - 1  y  1 sin   x    sin x
period: 360

34. Trigonometric Graphs plot boundary values
y
1 y  sin x
360 180 180 360 540 720 x
-1
domain : all real x odd function
range : - 1  y  1 sin   x    sin x
y period: 360
1
360 180 180 360 540 720 x
-1

35. Trigonometric Graphs plot boundary values
y
1 y  sin x
360 180 180 360 540 720 x
-1
domain : all real x odd function
range : - 1  y  1 sin   x    sin x
y period: 360
1
360 180 180 360 540 720 x
-1

36. Trigonometric Graphs plot boundary values
y
1 y  sin x
360 180 180 360 540 720 x
-1
domain : all real x odd function
range : - 1  y  1 sin   x    sin x
y period: 360
1
360 180 180 360 540 720 x
-1

37. Trigonometric Graphs plot boundary values
y
1 y  sin x
360 180 180 360 540 720 x
-1
domain : all real x odd function
range : - 1  y  1 sin   x    sin x
y period: 360
1 y  cos x
360 180 180 360 540 720 x
-1

38. Trigonometric Graphs plot boundary values
y
1 y  sin x
360 180 180 360 540 720 x
-1
domain : all real x odd function
range : - 1  y  1 sin   x    sin x
y period: 360
1 y  cos x
360 180 180 360 540 720 x
-1
domain : all real x
range : - 1  y  1
period: 360

39. Trigonometric Graphs plot boundary values
y
1 y  sin x
360 180 180 360 540 720 x
-1
domain : all real x odd function
range : - 1  y  1 sin   x    sin x
y period: 360
1 y  cos x
360 180 180 360 540 720 x
-1
domain : all real x even function
range : - 1  y  1 cos   x   cos x
period: 360

40. y
360 180 180 360 540 720 x

41. y
360 180 180 360 540 720 x

42. y y  tan x
360 180 180 360 540 720 x

43. y y  tan x
360 180 180 360 540 720 x

44. y y  tan x
360 180 180 360 540 720 x
domain: all real x except x  90  180k , where k is an integer
range: all real y
period: 180

45. y y  tan x
360 180 180 360 540 720 x
domain: all real x except x  90  180k , where k is an integer
range: all real y
 odd function
period: 180
tan   x    tan x

46. y y  tan x
360 180 180 360 540 720 x
domain: all real x except x  90  180k , where k is an integer
range: all real y
 odd function
period: 180 y
y  cosecx tan   x    tan x
1
360 180 180 360 x
-1

47. y y  tan x
360 180 180 360 540 720 x
domain: all real x except x  90  180k , where k is an integer
range: all real y
 odd function
period: 180 y
y  cosecx tan   x    tan x
1
360 180 180 360 x
-1

48. y y  tan x
360 180 180 360 540 720 x
domain: all real x except x  90  180k , where k is an integer
range: all real y
 odd function
period: 180 y
y  cosecx tan   x    tan x
1
360 180 180 360 x
-1

49. y y  tan x
360 180 180 360 540 720 x
domain: all real x except x  90  180k , where k is an integer
range: all real y
 odd function
period: 180 y
y  cosecx tan   x    tan x
1
360 180 180 360 x
-1

50. y y  tan x
360 180 180 360 540 720 x
domain: all real x except x  90  180k , where k is an integer
range: all real y
 odd function
period: 180 y
y  cosecx tan   x    tan x
1
360 180 180 360 x
-1

51. y y  tan x
360 180 180 360 540 720 x
domain: all real x except x  90  180k , where k is an integer
range: all real y
 odd function
period: 180 y
y  cosecx tan   x    tan x
1
Exercise 4D; 3ace etc,
360 180 180 360 x 4bdf etc, 5ace etc,
-1 6bdf, 7ace, 8b, 9bd,
10bd, 11ac,
13 ace etc, 14c, 16*