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Created by Mr. Lafferty Graphs of the form y = a sin x o Trigonometry Graphs www.mathsrevision.com Int 2 Graphs of the form y = a sin bx o Phase angle.

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Presentation on theme: "Created by Mr. Lafferty Graphs of the form y = a sin x o Trigonometry Graphs www.mathsrevision.com Int 2 Graphs of the form y = a sin bx o Phase angle."— Presentation transcript:

1 created by Mr. Lafferty Graphs of the form y = a sin x o Trigonometry Graphs Int 2 Graphs of the form y = a sin bx o Phase angle Solving Trig Equations Special trig relationships

2 created by Mr. Lafferty Starter Int 2

3 created by Mr. Lafferty Learning Intention Success Criteria 1.Identify the key points for various graphs. 1.To investigate graphs of the form y = a sin x o y = a cos x o y = tan x o Int 2 Sine Graph

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

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

6 created by Mr. Lafferty Sine Graph 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.

7 created by Mr. Lafferty Int 2 Sine Graph o 180 o 270 o 360 o y = 5sinx o y = 4sinx o y = sinx o y = -6sinx o

8 created by Mr. Lafferty Int 2 Cosine Graphs 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 Max value at x = 0 o and 360 o Minimum value at x = 180 o

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

10 created by Mr. Lafferty Int 2 Cosine Graph o 180 o 270 o 360 o y = cosx o y = 4cosx o y = 6cosx o y = cosx o y = -cosx o

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

12 created by Mr. Lafferty Int 2 Tangent Graphs

13 created by Mr. Lafferty Tangent Graph 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.

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

15 created by Mr. Lafferty Starter Int 2

16 created by Mr. Lafferty Learning Intention Success Criteria 1.Identify the key points for more complicated Trig graphs. 1.To investigate graphs of the form y = a sin bx o y = a cos bx o y = tan bx o Int 2 Trig Graphs

17 created by Mr. Lafferty Int 2 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 Let’s investigate the function y = sin bx

18 created by Mr. Lafferty Int 2 Sine Graph o 180 o 270 o 360 o y = sinx o y = sin2x o y = sin4x o y = sin0.5x o What effect does the number in front of x have on the graphs ?

19 created by Mr. Lafferty Int 2 Trigonometry Graphs y = a sin (bx) 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. How many times it repeats itself in 360 o

20 created by Mr. Lafferty Int 2 Cosine o 180 o 270 o 360 o y = cosx o y = cos2x o y = cos3x o What effect does the number at the front have on the graphs ?

21 created by Mr. Lafferty Int 2 Trigonometry Graphs y = a cos (bx) 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. How many times it repeats itself in 360 o

22 created by Mr. Lafferty Int 2 Trigonometry Graphs y = a tan (bx) 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. How many times it repeats itself in 180 o

23 y = 0.5sin2x o y = 2sin4x o y = 3sin0.5x o created by Mr. Lafferty Int 2 Trig Graph o 180 o 270 o 360 o Write down equations for graphs shown ? Combinations

24 created by Mr. Lafferty Int 2 Cosine o 180 o 270 o 360 o Write down equations for the graphs shown? Combinations y = 1.5cos2x o y = -2cos2x o y = 0.5cos4x o

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

26 created by Mr. Lafferty Starter Int 2

27 created by Mr. Lafferty Learning Intention Success Criteria 1.Understand the term phase angle / phase shift. 2.Read off the values for a and b for a graph of the form. y = a sin( x – c ) o 1.To explain what phase angle / phase shift is using knowledge from quadratics. Int 2 Phase Angle

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

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

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

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

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

33 created by Mr. Lafferty Int 2 Summary of work So far y = a sin (x - b) For b > 0 moves graph to the right along x – axis For b < 0 moves graph to the left along x – axis 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.

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

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

36 created by Mr. Lafferty Starter Int 2

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

38 created by Mr. Lafferty Int 2 Solving Trig Equations All +veSin +ve Tan +ve Cos +ve 180 o - x o 180 o + x o 360 o - x o 1234

39 created by Mr. Lafferty Int 2 Solving Trig Equations a sin x o + b = 0 Example 1 : Solving the equation sin x o = 0.5 in the range 0 o to 360 o Graphically what are we trying to solve x o = sin -1 (0.5) x o = 30 o There is another solution x o = 150 o (180 o – 30 o = 150 o ) sin x o = (0.5) 1234

40 created by Mr. Lafferty Int 2 Solving Trig Equations a sin x o + b = 0 Example 1 : Solving the equation 3sin x o + 1= 0 in the range 0 o to 360 o Graphically what are we trying to solve sin x o = -1/3 Calculate first Quad value x o = 19.5 o x = 180 o o = o ( 360 o o = o ) There is another solution 1234

41 created by Mr. Lafferty Int 2 Solving Trig Equations a cos x o + b = 0 Example 1 : Solving the equation cos x o = in the range 0 o to 360 o Graphically what are we trying to solve cos x o = x o = 51.3 o (360 o o = o ) x o = cos There is another solution 1234

42 created by Mr. Lafferty Int 2 Solving Trig Equations a tan x o + b = 0 Example 1 : Solving the equation tan x o = 2 in the range 0 o to 360 o Graphically what are we trying to solve tan x o = 2 x o = 63.4 o x = 180 o o = o x o = tan -1 (2) There is another solution 1234

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

44 created by Mr. Lafferty Starter Int 2

45 created by Mr. Lafferty Learning Intention Success Criteria 1.Know and learn the two special trig relationships. 2.Apply them to solve problems. 1.To explain some special trig relationships sin 2 x o + cos 2 x o = ? and tan x o and sin x cos x Int 2 Solving Trig Equations

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

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

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


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