1. Graphs of Trigonometric Functions

2. Graphs of Trigonometric Functions
This chapter focuses on using graphs of sinθ, cosθ and tanθ
We will be seeing how to work out values of these from the graphs
We are also going to look at transformations of these graphs

3. I teach using the Trig graphs so
haven’t included sections 8A and 8B on
CAST diagrams!

4. Teachings for Exercise 8C

5. Graphs of Trigonometric Functionsy
y = sinθ
-90º
-270º
-360º
-180º
You need to be able to recognise the graphs of sinθ, cosθ and tanθ
You will have seen all these graphs on your GCSE
The key points to remember are the peaks/troughs of each, and the points of intersection
The Cos graph is the same as the Sin graph, but shifted along (it starts at 1 instead of 0)
The Tan graph has lines called asymptotes. These are points the graph approaches but never reaches (90º, 270º etc…) 1 θ
90º
180º
270º
360º -1 y
y = cosθ 1 θ
-360º
-270º
-180º
-90º
90º
180º
270º
360º -1
y = tanθ 1 θ
-360º
-270º
-180º
-90º
90º
180º
270º
360º -1
Period (length of wave) = 360º for Sin and Cos, and 180º for Tan 8C

6. Graphs of Trigonometric Functionsy
y = sinθ
You need to be able to recognise the graphs of sinθ, cosθ and tanθ
These are the same graphs, but with radians instead… 1 θ
-360º
-2π
-270º
-3π 2
-180º -π
-90º
-π 2
π 2
90º
180º π
3π 2
270º
360º 2π -1 y
y = cosθ 1 θ
-360º
-2π
-270º
-3π 2
-180º -π
-90º
-π 2
π 2
90º
180º π
270º
3π 2
360º 2π -1
y = tanθ 1 θ
-360º
-2π
-270º
-3π 2
-180º -π
-90º
-π 2
π 2
90º
180º π
3π 2
270º
360º 2π -1 8C

7. Graphs of Trigonometric Functions
You need to be able to recognise the graphs of sinθ, cosθ and tanθ
You need to be able to work out larger values of sin, cos and tan as acute angles (0º - 90º)
 Write sin 130º as sine of an acute angle
(sometimes asked as a ‘trigonometric ratio’)
Sin 130º = Sin 50º y
y = sinθ
-40
-40 1 50
130 θ
90º
180º
270º
360º -1
 Draw a sketch of the graph
 Mark on 130º
 Using the fact that the graph has symmetry, find an acute value of θ which has the same value as sin 130 8C

8. Graphs of Trigonometric Functionsy
y = cosθ
You need to be able to recognise the graphs of sinθ, cosθ and tanθ
You need to be able to work out larger values of sin, cos and tan as acute angles (0º - 90º)
 Write cos (-120)º as cos of an acute angle
Cos(-120)º = -Cos 60º
+60
+60 1
-60 60
+30 θ
-270º
-180º
+30
-90º
90º
180º
270º
-120 -1
 Draw a sketch of the graph
 Mark on -120º
 Using the fact that the graph has symmetry, find an acute value of θ which has the same numerical value as cos (-120)
 The value you find here will have the same digits in it, but will be multiplied by -1 8C

9. Graphs of Trigonometric Functions
y = tanθ
You need to be able to recognise the graphs of sinθ, cosθ and tanθ
You need to be able to work out larger values of sin, cos and tan as acute angles (0º - 90º)
 Write tan 4π/3 as tan of an acute angle
Tan 4π/3 = Tan π/3 1
1 3
4 3 θ
1 3 +
π 2 π
1 3 +
3π 2 2π -1
 Draw a sketch of the graph
 Mark on 4π/3
 Using the fact that the graph has symmetry, find an acute value of θ which has the same numerical value as tan 4π/3 8C

10. Teachings for Exercise 8D

11. Graphs of Trigonometric Functions
You need to be able to find the exact values of some Trigonometrical Ratios
Some values of Sin, Cos or Tan can be written using fractions, surds, or combinations of both…
We can use an Equilateral Triangle with sides of length 2 to show this.
Using Pythagoras, the missing side in the right angled triangle is √3 (Square root of 22-12)
60˚ 2 2
60˚
60˚ 2
Hyp
30˚ 2
Opp √3
Opp Hyp
Sinθ =
Sin30 =
60˚ √
Sin60 = 1
Opp 8D

12. Graphs of Trigonometric Functions
You need to be able to find the exact values of some Trigonometrical Ratios
Some values of Sin, Cos or Tan can be written using fractions, surds, or combinations of both…
We can use an Equilateral Triangle with sides of length 2 to show this.
Using Pythagoras, the missing side in the right angled triangle is √3 (Square root of 22-12)
60˚ 2 2
60˚
60˚ 2
Hyp
30˚ 2
Adj √3
Adj Hyp √
Cosθ =
Cos30 =
60˚
Cos60 = 1
Adj 8D

13. Graphs of Trigonometric Functions
You need to be able to find the exact values of some Trigonometrical Ratios
Some values of Sin, Cos or Tan can be written using fractions, surds, or combinations of both…
We can use an Equilateral Triangle with sides of length 2 to show this.
Using Pythagoras, the missing side in the right angled triangle is √3 (Square root of 22-12)
60˚ 2 2
60˚
60˚ 2
30˚ 2
Adj
Opp √3
Opp Adj
1 √3 √ =
Tanθ =
Tan30 =
60˚
Tan60 = √3 1
Opp
Adj 8D

14. Graphs of Trigonometric Functions
You need to be able to find the exact values of some Trigonometrical Ratios
Some values of Sin, Cos or Tan can be written using fractions, surds, or combinations of both…
We can also do a similar demonstration with a right-angled Isosceles triangle, with the equal sides being of length 1 unit.
Using Pythagoras’ Theorem, the hypotenuse will be of length √2 (Square root of )
Hyp √2
Opp 1
45˚ 1
Opp Hyp
1 √2 √ =
Sinθ =
Sin45 = 8D

15. Graphs of Trigonometric Functions
You need to be able to find the exact values of some Trigonometrical Ratios
Some values of Sin, Cos or Tan can be written using fractions, surds, or combinations of both…
We can also do a similar demonstration with a right-angled Isosceles triangle, with the equal sides being of length 1 unit.
Using Pythagoras’ Theorem, the hypotenuse will be of length √2 (Square root of )
Hyp √2 1
45˚ 1
Adj
Adj Hyp
1 √2 √ =
Cosθ =
Cos45 = 8D

16. Graphs of Trigonometric Functions
You need to be able to find the exact values of some Trigonometrical Ratios
Some values of Sin, Cos or Tan can be written using fractions, surds, or combinations of both…
We can also do a similar demonstration with a right-angled Isosceles triangle, with the equal sides being of length 1 unit.
Using Pythagoras’ Theorem, the hypotenuse will be of length √2 (Square root of ) √2
Opp 1
45˚ 1
Adj
Opp Adj = 1
Tanθ =
Tan45 = 8D

17. 8E is just a reminder of the Trig
Graph shapes!

18. Teachings for Exercise 8F

19. Graphs of Trigonometric Functionsy
y = sinθ
You need to be able to recognise transformations of graphs, and sketch them
Transformation type 1
This stretches the graph vertically by a factor ‘a’.
“Multiplying sinθ by a number will affect the y value directly” 1 θ
90º
180º
270º
360º -1 y 3
y = 3sinθ θ
90º
180º
270º
360º
Y values 3 times as big -3
y = ½sinθ
Y values halved y
0.5 θ
90º
180º
270º
360º
-0.5 8F

20. Graphs of Trigonometric Functionsy
y = sinθ
You need to be able to recognise transformations of graphs, and sketch them
Transformation type 1
This stretches the graph vertically by a factor ‘a’. 1 θ
90º
180º
270º
360º -1 y
y = -sinθ 1 θ
90º
180º
270º
360º
Reflection in the x axis -1
(all the y values will ‘swap sign’) y
y = sin(-θ) 1
Reflection in the y axis θ
90º
180º
270º
360º
(You get the same y values for the reversed x value. -90 gives the result 90 would have) -1 8F

21. Graphs of Trigonometric Functionsy
y = cosθ
You need to be able to recognise transformations of graphs, and sketch them
Transformation type 1
This stretches the graph vertically by a factor ‘a’. 1 θ
90º
180º
270º
360º -1 y
y = -cosθ 1 θ
90º
180º
270º
360º
Reflection in the x axis -1 y
y = cos(-θ) 1
Reflection in the y axis θ
90º
180º
270º
360º -1 8F

22. Graphs of Trigonometric Functionsy
y = sinθ
You need to be able to recognise transformations of graphs, and sketch them
Transformation type 2
This shifts the graph vertically ‘a’ units.
 It is important to note that the ‘a’ is added on AFTER doing ‘sinθ’
“Adding an amount onto sinθ is a vertical shift” 1 θ
90º
180º
270º
360º -1 y
y = sinθ + 1 1 θ
90º
180º
270º
360º -1
Y values all increase by 1 y
y = -2 + sinθ -1
Y values all decrease by 2 θ -2 -3 8F

23. Graphs of Trigonometric Functionsy
y = sinθ
You need to be able to recognise transformations of graphs, and sketch them
Transformation type 3
This shifts the graph horizontally ‘-a’ units.
NOTE: The ‘a’ is added to θ before we work out the sine value…
“Adding/Subtracting an amount from the bracket is a horizontal shift” 1 θ
90º
180º
270º
360º -1 y 90
y = sin(θ + 90) 1 θ
90º
180º
270º
360º -1
Y takes the same set of values, for values of θ that are 90 less than before y 30
y = sin(θ – 30) 1
Y takes the same set of values, for values of θ that are 30 more than before θ
90º
180º
270º
360º -1 8F

24. Graphs of Trigonometric Functionsy
y = sinθ
You need to be able to recognise transformations of graphs, and sketch them
Transformation type 4
This stretches the graph horizontally by a factor ‘1/a’
“Multiplying or dividing θ in the bracket is a horizontal stretch/squash” 1 θ
90º
180º
270º
360º -1 y
y = sin2θ 1 θ
90º
180º
270º
360º
Same set of Y values, for half the θ values -1
Same set of y values, for triple the θ values y
y = sin(θ/3) 1 θ
270º
540º
810º
1080º -1 8F

25. Graphs of Trigonometric Functions
(90, 1.5) y
y = sinθ + k
You need to be able to answer questions with unknowns in
The graph shows the Function:
f(x) = Sinθ + k
a) Write down the value of k
 0.5 (Graph 0.5 units higher)
b) What is the smallest positive value of θ that gives a minimum point?
 270˚
c) What is the value of Sinθ at this point?
 -0.5 1 θ
90º
180º
270º
360º -1 8F

26. Graphs of Trigonometric Functionsy
y = cos(θ+k)
You need to be able to answer questions with unknowns in
The graph shows the Function:
f(x) = Cos(θ + k)
a) Write down the value of k
 20 (Graph moved 20 units left)
 f(x) = Cos(θ + 20)
b) What is the value of θ at x?
 x = 250˚
c) What are the coordinates of the minimum?
 (160, -1)
d) What is the value of Cosθ at y? 1 y
70º
250º xº θ -1
f(x) = Cos(θ + k)
We know k
f(x) = Cos(θ + 20)
On the y axis, θ = 0.
f(x) = Cos(20)
Work out the answer!
f(x) = 0.94 (2dp) 8F

27. Summary We have been reminded of the graphs for sine, cosine and tan
We have looked at finding equivalent values on these graphs
We have also looked at various graph transformations