Trigonometry Graphs Graphs of the form y = a sin xo Graphs of the form y = a sin bxo Phase angle Solving Trig Equations Special trig relationships

Sine Graph 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.

Sine Graph Key Features Zeros at 0, 180o and 360o Max value at x = 90o 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

What effect does the number at the front have on the graphs ? y = sinxo y = 2sinxo y = 3sinxo y = 0.5sinxo y = -sinxo What effect does the number at the front have on the graphs ? Sine Graph 3 2 1 90o 180o 270o 360o -1 -2 -3

Sine Graph 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.

Sine Graph 6 4 2 -2 -4 -6 y = 5sinxo y = 4sinxo y = sinxo y = -6sinxo 90o 180o 270o 360o -2 -4 -6

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

What effect does the number at the front have on the graphs ? y = cosxo y = 2cosxo y = 3cosxo y = 0.5cosxo y = -cosxo What effect does the number at the front have on the graphs ? Cosine 3 2 1 90o 180o 270o 360o -1 -2 -3

Cosine Graph 6 4 2 -2 -4 -6 y = cosxo y = 4cosxo y = 6cosxo y = -cosxo 90o 180o 270o 360o -2 -4 -6

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

Tangent Graphs created by Mr. Lafferty

Tangent Graph 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.

Trig Graphs Learning Intention Success Criteria To investigate graphs of the form y = a sin bxo y = a cos bxo y = tan bxo Identify the key points for more complicated Trig graphs.

Period of a Function y = sin bx 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

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

Trigonometry Graphs 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.

What effect does the number at the front have on the graphs ? Cosine y = cosxo y = cos2xo y = cos3xo Int 2 3 2 1 90o 180o 270o 360o -1 -2 -3

Trigonometry Graphs 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.

Trigonometry Graphs 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.

Write down equations for graphs shown ? y = 0.5sin2xo y = 2sin4xo y = 3sin0.5xo Write down equations for graphs shown ? Trig Graph Combinations 3 2 1 90o 180o 270o 360o -1 -2 -3

Write down equations for the graphs shown? Cosine y = 1.5cos2xo y = -2cos2xo y = 0.5cos4xo Combinations 3 2 1 90o 180o 270o 360o -1 -2 -3

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

Sine Graph y = sin(x - 45)o 1 To the right “-” -1 By how much do we have to move the standard sine curve so it fits on the other sine curve? Sine Graph y = sin(x - 45)o 1 To the right “-” 45o 45o 90o 180o 270o 360o -1

Sine Graph y = sin(x + 60)o 1 To the left “+” -1 By how much do we have to move the standard sine curve so it fits on the other sine curve? Sine Graph y = sin(x + 60)o 1 To the left “+” 60o -60o 90o 180o 270o 360o -1

Phase Angle y = sin (x - c) Moves graph along x - axis For c > 0 moves graph to the right along x – axis For c < 0 moves graph to the left along x – axis

Cosine Graph y = cos(x - 70)o 1 To the right “-” -1 By how much do we have to move the standard cosine curve so it fits on the other cosine curve? Cosine Graph y = cos(x - 70)o 1 To the right “-” 70o 90o 160o 180o 270o 360o -1

Cosine Graph y = cos(x + 56)o 1 To the left “+” -1 By how much do we have to move the standard cosine curve so it fits on the other cosine curve? Cosine Graph y = cos(x + 56)o 1 To the left “+” 56o 34o 90o 180o 270o 360o -1

y = a sin (x - b) Summary of work So far For a > 1 stretches graph in the y-axis direction For b > 0 moves graph to the right along x – axis For a < 1 compresses graph in the y - axis direction For b < 0 moves graph to the left along x – axis For a - negative flips graph in the x – axis.

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

Solving Trig Equations 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.

Solving Trig Equations 1 2 3 4 Sin +ve All +ve 180o - xo 180o + xo 360o - xo Tan +ve Cos +ve

Solving Trig Equations Graphically what are we trying to solve a sin xo + b = 0 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)

Solving Trig Equations Graphically what are we trying to solve a sin xo + b = 0 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 + 19.5o = 199.5o There is another solution ( 360o - 19.5o = 340.5o)

Solving Trig Equations Graphically what are we trying to solve a cos xo + b = 0 Example 1 : Solving the equation cos xo = 0.625 in the range 0o to 360o 1 2 3 4 cos xo = 0.625 xo = cos -1 0.625 xo = 51.3o There is another solution (360o - 53.1o = 308.7o)

Solving Trig Equations Graphically what are we trying to solve a tan xo + b = 0 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 + 63.4o = 243.4o

Solving Trig Equations 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.

Solving Trig Equations Lets investigate sin 2xo + cos 2 xo = ? Calculate value for x = 10, 20, 50, 250 sin 2xo + cos 2 xo = 1 Learn !

Solving Trig Equations Lets investigate sin xo cos xo tan xo and Calculate value for x = 10, 20, 50, 250 sin xo cos xo tan xo = Learn !