1. Trigonometric Functions
Sine and Cosine Functions

2. f(x) = sin x and f(x) = cos x

3. f(x) = sin x and f(x) = cos x

4. f(x) = sin x & two important ideas
Period
Amplitude
Amplitude
Period
Period means how many degrees in one cycle.
Amplitude means the distance from the centre to the maximum or minimum, OR (max + min) ÷ 2

5. f(x) = sin x
Period = 360º
Amplitude = 1
Period

6. Now we will investigate
f(x) = A sin Bx + C
How do A, B and C affect the shape of the graph?
Note: It is exactly the same for sine and cosine, so we will stick just to sine for the start.

7. f(x) = sin x & f(x) = sin x + 3

8. f(x) = sin x + 3 & f(x) = sin x – 2
So C moves the curve up and down

9. f(x) = sin x & f(x) = sin 2x
Period = 180º

10. f(x) = sin x & f(x) = sin 3x So B changes the period;
= 120º
So B changes the period;
the period of the function is (360º ÷ B)

11. f(x) = sin x & f(x) = 2 sin x
Amplitude = 2

12. f(x) = sin x & f(x) = 4 sin x
Amplitude = 4

13. f(x) = sin x & f(x) = -1 sin x
Amplitude = 1

14. f(x) = sin x & f(x) = -3 sin x
Amplitude = 3

15. f(x) = sin x & f(x) = A sin x
The A gives the amplitude of the function.
A negative value means the graph goes down – up, not up – down.

16. f(x) = A sin Bx + C A = amplitude B = 360º ÷ period C = vertical shift
Note: It is exactly the same for sine and cosine.
The difference is the where it crosses the y-axis.

17. What is the equation of this function?
Amplitude = 2
so, A = 2
so, B = 3
Period = 120º
so, C = -1
Vertical shift = -1
f(x) = 2 sin 3x – 1

18. What is the equation of this function?
Amplitude = 4,
going down-up
so, A = -4
Period = 720º
so, B = 0.5
Vertical shift = 1
so, C = 1
f(x) = -4 sin ½x + 1

19. What is the equation of this function?
Amplitude = 2.5
so, A = 2.5
Period = 240º
so, B = 1.5
Vertical shift = 2
so, C = 2
f(x) = 2.5 sin 1.5x + 2