1. Graphs of the form y = a sin x o Nat 5 Graphs of the form y = a sin bx o Phase angle Solving Trig Equations Special trig relationships Trigonometric Functions and Graphs

2. Learning Intention 1.To identify key features of graphs of trigonometric functions including: y = a sin x o y = a cos x o y = a tan x o Trigonometric Functions and Graphs

3. Sine Function Graph Key Features Key Features Domain is 0 to 360 o (repeats itself every 360 o ) Maximum value of 1 Minimum value of -1 Zeros at 0, 180 o and 360 o Maximum value at x = 90 o Minimum value at x = 270 o

4. 1 2 3 -3 -2 0 90 o 180 o 270 o 360 o Sine Function Graph y = sinx o y = 2sinx o y = 3sinx o y = 0.5sinx o y = -sinx o

5. 2 4 6 -6 -4 -2 0 90 o 180 o 270 o 360 o y = 5sinx o y = 4sinx o y = sinx o y = -6sinx o

6. Cosine Function Graph Key Features Key Features Domain is 0 to 360 o (repeats itself every 360 o ) Maximum value of 1 Minimum value of -1 Zeros at 90 o and 270 o Maximum value at x = 0 o and 360 o Minimum value at x = 180 o

7. 1 2 3 -3 -2 0 90 o 180 o 270 o 360 o y = cosx o y = 2cosx o y = 3cosx o y = 0.5cosx o y = -cosx o

8. Cosine Function Graph 2 4 6 -6 -4 -2 0 90 o 180 o 270 o 360 o y = cosx o y = 4cosx o y = 6cosx o y = 0.5cosx o y = -1.5cosx o

9. Tangent Function Graph Key Features Domain is 0 to 180 o (repeats itself every 180 o ) Key Features Key Features Zeros at 0 and 180 o Undefined at 90 o and 270 o

10. created by Mr. Lafferty Tangent Function Graph

11. y = a cos (x) For a > 1 stretches graph in the y-axis direction. For a < 1 compresses graph in the y - axis direction. For a < 0 graph reflects in the x – axis. Cosine Function Graph y = a sin (x) y = a tan (x)

12. Learning Intention 1.To identify key features of graphs of trigonometric functions including: y = sin bx o y = cos bx o y = tan bx o Trigonometric Functions and Graphs

13. When a pattern repeats itself over and over, it is said to be periodic. Sine function has a period of 360 o Period of a Function Consider y = sin bx Cosine function has a period of 360 o y = cos bx and

14. Sine Function Graph 1 2 3 -3 -2 0 90 o 180 o 270 o 360 o y = sinx o y = sin2x o y = sin4x o y = sin0.5x o

15. Cosine Function Graph 1 2 3 -3 -2 0 90 o 180 o 270 o 360 o y = cosx o y = cos2x o y = cos3x o

16. When a pattern repeats itself over and over, it is said to be periodic. Tangent function has a period of 180 o Period of a Function Consider y = tan bx

17. Tangent Function Graph y = tanx o

18. Tangent Function Graph y = tan2x o

19. Tangent Function Graph y = tan3x o

20. y = a sin (bx) b is how many times graph repeats itself in 360 o Sine, Cosine & Tangent Functions y = a cos (bx) y = a tan (bx) b is how many times it repeats itself in 180 o

21. Learning Intentions 1.To identify key features of graphs of trigonometric functions including: y = asin bx o y = acos bx o y = atan bx o Trigonometric Functions and Graphs 2.To sketch graphs of trigonometric functions of this form.

22. Trigonometric Graphs 1 2 3 -3 -2 0 90 o 180 o 270 o 360 o Write down an equation for the graph shown. y = 0.5sin2x o y = 2sin4x o y = 3sin0.5x o

23. Trigonometric Graphs 1 2 3 -3 -2 0 90 o 180 o 270 o 360 o y = 1.5cos2x o y = -2cos2x o y = 0.5cos4x o Write down an equation for the graph shown.

24. Learning Intentions 1.To identify the phase angle in graphs of trigonometric functions of the form: y = a sin (x-b) o Trigonometric Functions and Graphs 2.To sketch graphs of trigonometric functions of the form: y = a sin (x-b) o

25. The Sine Function Graph 1 0 90 o 180 o 270 o 360 o y = sin(x - 45) o 45 o 45 o

26. The Sine Function Graph 1 0 90 o 180 o 270 o 360 o -60 o y = sin(x + 60) o 60 o

27. The Cosine Function Graph 70 o Int 2 1 0 90 o 180 o 270 o 360 o y = cos(x - 70) o 160 o By how much do we have to move the ‘new’ cosine curve so it fits on the original cosx o curve?

28. 56 o Int 2 1 0 90 o 180 o 270 o 360 o y = cos(x + 56) o 34 o The Cosine Function Graph By how much do we have to move the ‘new’ cosine curve so it fits on the original cosx o curve?

29. Phase Angle y = sin (x - b) b moves graph along x – axis. y = cos (x - b)

30. Naming a Function y = a cos (x – b) a =3 b = -30 y = 3 cos (x - 30)

31. Phase Angle & Graphs 1 0 180 o 360 o 540 o 720 o By how much do we have to move the cosx o curve so it fits exactly onto the sinx o curve? cos(x+90) o = sinx o Similarly, sin(x-90) o = cosx o sinx o and cosx o are 90 o out of phase.