1. Trigonometry Graphs www.mathsrevision.com
Int 2
Graphs of the form y = a sin xo
Graphs of the form y = a sin bxo
Phase angle
Solving Trig Equations
Special trig relationships
created by Mr. Lafferty

2. Starter
Int 2
created by Mr. Lafferty

3. Sine Graph www.mathsrevision.com Int 2 Learning Intention
Success Criteria
To investigate graphs of the form
y = a sin xo
y = a cos xo
y = tan xo
Identify the key points for various graphs.
created by Mr. Lafferty

4. Sine Graph www.mathsrevision.com Key Features
Zeros at 0, 180o and 360o
Max value at x = 90o
Int 2
Minimum value at x = 270o
Key Features
Domain is 0 to 360o
(repeats itself every 360o)
Maximum value of 1
Minimum value of -1
created by Mr. Lafferty

5. What effect does the number at the front have on the graphs ?
y = sinxo
y = 2sinxo
y = 3sinxo
y = 0.5sinxo
y = -sinxo
Sine Graph
Int 2 3 2 1
90o
180o
270o
360o -1 -2 -3
created by Mr. Lafferty

6. Sine Graph y = a sin (x) www.mathsrevision.com
Int 2
y = a sin (x)
For a > 1 stretches graph in the y-axis direction
For a < 1 compresses graph in the y - axis direction
For a - negative flips graph in the x – axis.
created by Mr. Lafferty

7. Sine Graph www.mathsrevision.com 6 4 2 -2 -4 -6 y = 5sinxo y = 4sinxo
Int 2 6 4 2
90o
180o
270o
360o -2 -4 -6
created by Mr. Lafferty

8. Cosine Graphs www.mathsrevision.com Key Features Zeros at 90o and 270o
Max value at x = 0o and 360o
Int 2
Minimum value at x = 180o
Key Features
Domain is 0 to 360o
(repeats itself every 360o)
Maximum value of 1
Minimum value of -1
created by Mr. Lafferty

9. What effect does the number at the front have on the graphs ?
y = cosxo
y = 2cosxo
y = 3cosxo
y = 0.5cosxo
y = -cosxo
Cosine
Int 2 3 2 1
90o
180o
270o
360o -1 -2 -3
created by Mr. Lafferty

10. Cosine Graph www.mathsrevision.com 6 4 2 -2 -4 -6 y = cosxo y = 4cosxo
Int 2 6 4 2
90o
180o
270o
360o -2 -4 -6
created by Mr. Lafferty

11. Tangent Graphs www.mathsrevision.com Key Features Zeros at 0 and 180o
Int 2
Key Features
Domain is 0 to 180o
(repeats itself every 180o)
created by Mr. Lafferty

12. Tangent Graphs
Int 2
created by Mr. Lafferty

13. Tangent Graph y = a tan (x) www.mathsrevision.com
Int 2
y = a tan (x)
For a > 1 stretches graph in the y-axis direction
For a < 1 compresses graph in the y - axis direction
For a - negative flips graph in the x – axis.
created by Mr. Lafferty

14. Combination Graphs Revision Booklet All questions
Int 2
Revision Booklet
All questions
created by Mr. Lafferty

15. Starter
Int 2
created by Mr. Lafferty

16. Trig Graphs www.mathsrevision.com Int 2 Learning Intention
Success Criteria
To investigate graphs of the form
y = a sin bxo
y = a cos bxo
y = tan bxo
Identify the key points for more complicated Trig graphs.
created by Mr. Lafferty

17. Period of a Function y = sin bx www.mathsrevision.com
Int 2
When a pattern repeats itself over and over,
it is said to be periodic.
Sine function has a period of 360o
Let’s investigate the function
y = sin bx
created by Mr. Lafferty

18. What effect does the number in front of x have on the graphs ?
y = sinxo
y = sin2xo
y = sin4xo
y = sin0.5xo
Sine Graph
Int 2 3 2 1
90o
180o
270o
360o -1 -2 -3
created by Mr. Lafferty

19. Trigonometry Graphs y = a sin (bx) www.mathsrevision.com
Int 2
y = a sin (bx)
How many times
it repeats
itself in 360o
For a > 1 stretches graph in the y-axis direction
For a < 1 compresses graph in the y - axis direction
For a - negative flips graph in the x – axis.
created by Mr. Lafferty

20. What effect does the number at the front have on the graphs ?
Cosine
y = cosxo
y = cos2xo
y = cos3xo
Int 2 3 2 1
90o
180o
270o
360o -1 -2 -3
created by Mr. Lafferty

21. Trigonometry Graphs y = a cos (bx) www.mathsrevision.com
Int 2
y = a cos (bx)
How many times
it repeats
itself in 360o
For a > 1 stretches graph in the y-axis direction
For a < 1 compresses graph in the y - axis direction
For a - negative flips graph in the x – axis.
created by Mr. Lafferty

22. Trigonometry Graphs y = a tan (bx) www.mathsrevision.com
Int 2
y = a tan (bx)
How many times
it repeats
itself in 180o
For a > 1 stretches graph in the y-axis direction
For a < 1 compresses graph in the y - axis direction
For a - negative flips graph in the x – axis.
created by Mr. Lafferty

23. Write down equations for graphs shown ?
y = 0.5sin2xo
y = 2sin4xo
y = 3sin0.5xo
Trig Graph
Combinations
Int 2 3 2 1
90o
180o
270o
360o -1 -2 -3
created by Mr. Lafferty

24. Write down equations for the graphs shown?
y = 1.5cos2xo
y = -2cos2xo
y = 0.5cos4xo
Cosine
Combinations
Int 2 3 2 1
90o
180o
270o
360o -1 -2 -3
created by Mr. Lafferty

25. Combination Graphs Revision Booklet All questions
Int 2
Revision Booklet
All questions
created by Mr. Lafferty

26. Starter
Int 2
created by Mr. Lafferty

27. Phase Angle www.mathsrevision.com Int 2 Learning Intention
Success Criteria
To explain what phase angle / phase shift is using knowledge from quadratics.
Understand the term phase angle / phase shift.
Read off the values
for a and b for a graph of the form.
y = a sin( x – c )o
created by Mr. Lafferty

28. Sine Graph www.mathsrevision.com y = sin(x - 45)o 1 To the right “-”
By how much do we have to move the standard sine curve so it fits on the other sine curve?
Sine Graph
Int 2
y = sin(x - 45)o 1
To the right “-”
45o
45o
90o
180o
270o
360o -1
created by Mr. Lafferty

29. Sine Graph www.mathsrevision.com y = sin(x + 60)o 1 To the left “+” -1
By how much do we have to move the standard sine curve so it fits on the other sine curve?
Sine Graph
Int 2
y = sin(x + 60)o 1
To the left “+”
60o
-60o
90o
180o
270o
360o -1
created by Mr. Lafferty

30. Phase Angle y = sin (x - c) www.mathsrevision.com Moves graph
Int 2
y = sin (x - c)
Moves graph
along x - axis
For c > 0 moves graph to the right along x – axis
For c < 0 moves graph to the left along x – axis
created by Mr. Lafferty

31. Cosine Graph www.mathsrevision.com y = cos(x - 70)o 1 To the right “-”
By how much do we have to move the standard cosine curve so it fits on the other cosine curve?
Cosine Graph
Int 2
y = cos(x - 70)o 1
To the right “-”
70o
90o
160o
180o
270o
360o -1
created by Mr. Lafferty

32. Cosine Graph www.mathsrevision.com y = cos(x + 56)o 1 To the left “+”
By how much do we have to move the standard cosine curve so it fits on the other cosine curve?
Cosine Graph
Int 2
y = cos(x + 56)o 1
To the left “+”
56o
34o
90o
180o
270o
360o -1
created by Mr. Lafferty

33. y = a sin (x - b) Summary of work So far www.mathsrevision.com
Int 2
y = a sin (x - b)
For a > 1 stretches graph
in the y-axis direction
For b > 0 moves graph to
the right along x – axis
For a < 1 compresses graph
in the y - axis direction
For b < 0 moves graph to
the left along x – axis
For a - negative flips graph
in the x – axis.
created by Mr. Lafferty

34. Sketch Graph www.mathsrevision.com y = a cos (x – b) a =3 b =30
Int 2
a =3
b =30
y = 2 cos (x - 30)
created by Mr. Lafferty

35. Combination Graphs Revision Booklet All questions
Int 2
Revision Booklet
All questions
created by Mr. Lafferty

36. Starter
Int 2
created by Mr. Lafferty

37. Solving Trig Equations
Int 2
Learning Intention
Success Criteria
To explain how to solve
trig equations of the form
a sin xo + 1 = 0
Use the rule for solving any ‘ normal ‘ equation
Realise that there are many solutions to trig equations depending on domain.
created by Mr. Lafferty

38. Solving Trig Equations
Int 2 1 2 3 4
Sin +ve
All +ve
180o - xo
180o + xo
360o - xo
Tan +ve
Cos +ve
created by Mr. Lafferty

39. Solving Trig Equations
Graphically what are we trying to solve
a sin xo + b = 0
Int 2
Example 1 :
Solving the equation sin xo = 0.5 in the range 0o to 360o
sin xo = (0.5) 1 2 3 4
xo = sin-1(0.5)
xo = 30o
There is another solution
xo = 150o
(180o – 30o = 150o)
created by Mr. Lafferty

40. Solving Trig Equations
Graphically what are we trying to solve
a sin xo + b = 0
Int 2
Example 1 :
Solving the equation 3sin xo + 1= 0 in the range 0o to 360o 1 2 3 4
sin xo = -1/3
Calculate first Quad value
xo = 19.5o
x = 180o o = 199.5o
There is another solution
( 360o o = 340.5o)
created by Mr. Lafferty

41. Solving Trig Equations
Graphically what are we trying to solve
a cos xo + b = 0
Int 2
Example 1 :
Solving the equation cos xo = in the range 0o to 360o 1 2 3 4
cos xo = 0.625
xo = cos
xo = 51.3o
There is another solution
(360o o = 308.7o)
created by Mr. Lafferty

42. Solving Trig Equations
Graphically what are we trying to solve
a tan xo + b = 0
Int 2
Example 1 :
Solving the equation tan xo = 2 in the range 0o to 360o 1 2 3 4
tan xo = 2
xo = tan -1(2)
xo = 63.4o
There is another solution
x = 180o o = 243.4o
created by Mr. Lafferty

43. Solving Trig Equations
Int 2
Now try MIA Ex6
First Column Only
(page 249)
created by Mr. Lafferty

44. Starter
Int 2
created by Mr. Lafferty

45. Solving Trig Equations
Int 2
Learning Intention
Success Criteria
To explain some special trig relationships
sin 2 xo + cos 2 xo = ?
and
tan xo and sin x
cos x
Know and learn the two special trig relationships.
Apply them to solve problems.
created by Mr. Lafferty

46. Solving Trig Equations
Int 2
Lets investigate
sin 2xo + cos 2 xo = ?
Calculate value for x = 10, 20, 50, 250
sin 2xo + cos 2 xo = 1
Learn !
created by Mr. Lafferty

47. Solving Trig Equations
Int 2
Lets investigate
sin xo
cos xo
tan xo
and
Calculate value for x = 10, 20, 50, 250
sin xo
cos xo
tan xo =
Learn !
created by Mr. Lafferty

48. Solving Trig Equations
Int 2
Now try MIA Ex7
(page 252)
created by Mr. Lafferty