1. Trigonometry Graphs www.mathsrevision.com
S4 Credit
Exact values for Sin Cos and Tan
Angles greater than 90o
Graphs of the form y = a sin xo
Graphs of the form y = a sin bxo
Solving Trig Equations
Special trig relationships
created by Mr. Lafferty

2. Created by Mr Lafferty Maths Dept
Starter Questions
S4 Credit
28-May-18
Created by Mr Lafferty Maths Dept

3. Created by Mr Lafferty Maths Dept
Exact Values
S4 Credit
Learning Intention
Success Criteria
To build on basic trigonometry values.
Recognise basic triangles and exact values for sin, cos and tan 30o, 45o, 60o .
Calculate exact values for problems.
28-May-18
Created by Mr Lafferty Maths Dept

4. This triangle will provide exact values for
S4 Credit
Some special values of Sin, Cos and Tan are useful left as fractions, We call these exact values
60º 2 2
60º
30º 3 1
This triangle will provide exact values for
sin, cos and tan 30º and 60º

5. Exact Values ½  ½ 1 1 x 0º 30º 45º 60º 90º www.mathsrevision.com 3
S4 Credit x 0º
30º
45º
60º
90º
Sin xº
Cos xº
Tan xº  3 2 1 3 2 1 3

6. Exact Values 45º 2 1 1 45º 1 1 www.mathsrevision.com
S4 Credit
Consider the square with sides 1 unit
45º 2 1 1
45º 1 1
We are now in a position to calculate
exact values for sin, cos and tan of 45o

7. Exact Values ½  ½ 1 1 1 x 0º 30º 45º 60º 90º 1 2
S5 Int2 x 0º
30º
45º
60º
90º
Sin xº
Cos xº
Tan xº  3 2
1 2 1 3 2
1 2 1 1 3

8. Created by Mr Lafferty Maths Dept
Exact Values
S5 Int2
Now try Ex 2.1
Ch11 (page 220)
28-May-18
Created by Mr Lafferty Maths Dept

9. Created by Mr Lafferty Maths Dept
Starter Questions
S4 credit
28-May-18
Created by Mr Lafferty Maths Dept

10. Created by Mr. Lafferty Maths Dept.
Angles Greater than 90o
S4 credit
Learning Intention
Success Criteria
Introduce definition of sine, cosine and tangent over 360o using triangles with the unity circle.
Find values of sine, cosine and tangent over the range 0o to 360o.
2. Recognise the symmetry and equal values for sine, cosine and tangent.
28-May-18
Created by Mr. Lafferty Maths Dept.

11. Angles Greater than 90o www.mathsrevision.com r r y x
5/28/2018 x y r
Angles Greater than 90o
S4 credit
We will now use a new definition to cater for ALL angles.
New Definitions
y-axis y x
P(x,y) r
We start by find the equation of a circle centre the origin.
First draw set axises x,y and then label the origin O.
Next we plot a point P say, which as coordinates x,y.
Next draw a line from the origin O to the point P and label
length of this line r.
If we now rotate the point P through 360 degrees keep the Origin fixed we trace out a circle with radius r and centre O.
Remembering Pythagoras’s Theorem from Standard grade
a square plus b squared equal c squares we can now write down the equal of any circle with centre the origin. Ao O
x-axis
28-May-18

12. Trigonometry www.mathsrevision.com 53o Angles over 900
S4 credit
Example
The radius line is 2cm.
The point (1.2, 1.6).
Find sin cos and tan for
the angle.
(1.2, 1.6)
Check answer with calculator
53o
28-May-18
Created by Mr Lafferty Maths Dept

13. Trigonometry www.mathsrevision.com 127o Angles over 900 Example 1
S4 credit
Check answer with calculator
The radius line is 2cm.
The point (-1.8, 0.8).
Find sin cos and tan for
the angle.
(-1.8, 0.8)
127o
28-May-18
Created by Mr Lafferty Maths Dept

14. Created by Mr Lafferty Maths Dept
Summary of results
Trigonometry
All Quadrants
Example
S4 credit
Calculate the ration for sin cos and tan
for the angle values below.
90o
30o
210o
45o
225o
Sin +ve
All +ve
60o
240o
180o - xo xo
120o
300o
180o 0o
135o
315o
180o + xo
360o - xo
150o
330o
Tan +ve
Cos +ve
Sin x
Cos x
Tan x
28-May-18
Created by Mr Lafferty Maths Dept
270o

15. What Goes In The Box ? www.mathsrevision.com
S4 credit
Write down the equivalent values of the following in term of the first quadrant (between 0o and 90o):
Sin 135o
Cos 150o
Tan 135o
Sin 225o
Cos 270o
Sin 300o
Cos 360o
Tan 330o
Sin 380o
Cos 460o
sin 45o
- sin 60o
-cos 45o
cos 0o
-tan 45o
- tan 30o
-sin 45o
sin 20o
-cos 90o
- cos 80o

16. Now try MIA Ch11 Ex3.1 Ch11 (page 222)
Trigonometry
Angles over 900
S4 credit
Now try MIA Ch11 Ex3.1 Ch11 (page 222)
28-May-18
Created by Mr Lafferty Maths Dept

17. Starter
S4 Credit
created by Mr. Lafferty

18. Sine Graph www.mathsrevision.com Learning Intention Success Criteria
S4 Credit
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

19. Sine Graph www.mathsrevision.com Key Features
Zeros at 0, 180o and 360o
Max value at x = 90o
S4 Credit
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

20. Sine Graph www.mathsrevision.com 3 2 1 -1 -2 -3 y = sinxo y = 2sinxo
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
S4 Credit 3 2 1
90o
180o
270o
360o -1
What effect does the negative sign have on the graphs ? -2 -3
created by Mr. Lafferty

21. Sine Graph y = a sin (x) www.mathsrevision.com
S4 Credit
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

22. Sine Graph www.mathsrevision.com 6 4 2 -2 -4 -6 y = 5sinxo y = 4sinxo
S4 Credit 6 4 2
90o
180o
270o
360o -2 -4 -6
created by Mr. Lafferty

23. Cosine Graphs www.mathsrevision.com Key Features
Zeros at 0, 90o and 270o
Max value at x = 0o and 360o
S4 Credit
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

24. What effect does the number at the front have on the graphs ?
y = cosxo
y = 2cosxo
y = 3cosxo
y = 0.5cosxo
y = -cosxo
Cosine
S4 Credit 3 2 1
90o
180o
270o
360o -1 -2 -3
created by Mr. Lafferty

25. Cosine Graph www.mathsrevision.com 6 4 2 -2 -4 -6 y = 2cosxo
S4 Credit 6 4 2
90o
180o
270o
360o -2 -4 -6
created by Mr. Lafferty

26. Tangent Graphs www.mathsrevision.com Key Features Zeros at 0 and 180o
S4 Credit
Key Features
Domain is 0 to 180o
(repeats itself every 180o)
created by Mr. Lafferty

27. Tangent Graphs
S4 Credit
created by Mr. Lafferty

28. Tangent Graph y = a tan (x) www.mathsrevision.com
S4 Credit
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

29. Period of a Function y = sin bx www.mathsrevision.com
S4 Credit
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

30. What effect does the number in front of x have on the graphs ?
y = sinxo
y = sin2xo
y = sin4xo
y = sin0.5xo
Sine Graph
S4 Credit 3 2 1
90o
180o
270o
360o -1 -2 -3
created by Mr. Lafferty

31. Trigonometry Graphs y = a sin (bx) www.mathsrevision.com
S4 Credit
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

32. Cosine www.mathsrevision.com 3 2 1 -1 -2 -3 y = cosxo y = cos2xo
S4 Credit 3 2 1
90o
180o
270o
360o -1 -2 -3
created by Mr. Lafferty

33. Trigonometry Graphs y = a cos (bx) www.mathsrevision.com
S4 Credit
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

34. Trigonometry Graphs y = a tan (bx) www.mathsrevision.com
S4 Credit
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

35. Write down the equations for the graphs shown ?
y = 0.5sin2xo
y = 2sin4xo
y = -3sin0.5xo
Trig Graph
Combinations
S4 Credit 3 2 1
90o
180o
270o
360o -1 -2 -3
created by Mr. Lafferty

36. Write down equations for the graphs shown?
y = 1.5cos2xo
y = -2cos2xo
y = 0.5cos4xo
Cosine
Combinations
S4 Credit 3 2 1
90o
180o
270o
360o -1 -2 -3
created by Mr. Lafferty

37. Combination Graphs Now Try MIA Ch11 Ex 5.1 Page 227
S4 Credit
Now Try MIA Ch11 Ex 5.1
Page 227
created by Mr. Lafferty

38. Sine Graph www.mathsrevision.com Simply move graph up by 1 1 -1 45o
S4 Credit 1
45o
90o
180o
270o
360o
Given the basic y = sin x graph what does the graph of y = sin x +1 look like? -1
created by Mr. Lafferty

39. Cosine Graph www.mathsrevision.com Simply move down by 0.5 1 -1
Given the y = cos x graph. What does the graph of y = cos x – 0.5 look like?
Cosine Graph
Simply move down by 0.5
S4 Credit 1
90o
160o
180o
270o
360o -1
created by Mr. Lafferty

40. Write down equations for graphs shown ?
y = 0.5sin2xo + 1
y = 2sin4xo- 1
Trig Graph
Combinations
S4 Credit 3 2 1
90o
180o
270o
360o -1 -2 -3
created by Mr. Lafferty

41. Write down equations for the graphs shown?
Cosine
y = cos2xo + 1
y = -2cos2xo - 1
Combinations
S4 Credit 3 2 1
90o
180o
270o
360o -1 -2 -3
created by Mr. Lafferty

42. Combination Graphs Now try MIA Ch11 Ex 5.2 Page 227
S4 Credit
Now try MIA Ch11 Ex 5.2
Page 227
created by Mr. Lafferty

43. Starter
S4 Credit
created by Mr. Lafferty

44. Solving Trig Equations
S4 Credit
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

45. Solving Trig Equations
S4 Credit 1 2 3 4
Sin +ve
All +ve
180o - xo
180o + xo
360o - xo
Tan +ve
Cos +ve
created by Mr. Lafferty

46. Solving Trig Equations
Graphically what are we trying to solve
a sin xo + b = 0
S4 Credit
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

47. Solving Trig Equations
Graphically what are we trying to solve
a sin xo + b = 0
S4 Credit
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

48. Solving Trig Equations
Graphically what are we trying to solve
a cos xo + b = 0
S4 Credit
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

49. Solving Trig Equations
Graphically what are we trying to solve
a tan xo + b = 0
S4 Credit
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

50. Solving Trig Equations
S4 Credit
Now try MIA Ch11
Ex6.1, 6.2 and 7.1
(page 236)
created by Mr. Lafferty

51. Starter
S4 Credit
created by Mr. Lafferty

52. Solving Trig Equations
S4 Credit
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

53. Solving Trig Equations
S4 Credit
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

54. Solving Trig Equations
S4 Credit
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

55. Solving Trig Equations
S4 Credit
Now try MIA Ex8.1
Ch11 (page 238)
created by Mr. Lafferty