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© Teachers Teaching with Technology (Scotland)

©Teachers Teaching with Technology (Scotland)

Teachers Teaching with Technology

T3 Scotland

Wave Function

T3 Scotland Wave Function Page 1 of 12

WAVE FUNCTION

AimThe aim of this module is to investigate trigonometric functions of the form

a x b x k (x a )cos sin cos+ = −

Objectives

Mathematical objectivesBy the end of this session you should be able to

• Know and describe the effect of combining two trig. graphs.• be able to write a function of the form acos(x) + bsin(x) in the form kcos(x - α)

Calculator objectivesBy the end of this session you should be able to

• Graph functions via [Y=].• Draw graphs of trigonometric functions using appropriate settings, in different graph

types.• Output a table of values from the TI-83• Know how to use the [VARS] button to define a function in terms of one already

defined.

� T3 Scotland Wave Function of 12

Calculator Skills SheetBefore we can start on this topic we must first ensure that your TI 83 is in the correct MODE, and is going to operate as we want it to.This is how we do this

2. We want to set a window range on the calculator to do this

press WINDOW

and using the cursor key change each line so that your screen looks like this.

Remember to press ENTER to confirm each line selection.

1. Press the MODE button.The display should look exactly like this.If it does not look like this, then using the cursor keys

highlight the correct item in each line and press ENTER to change the selection.Notice: There can only be one item in each line highlighted.

3. Press the 2 nd and FORMAT .This takes you to the WINDOW FORMAT screen.It should look like this.If it does not then using the cursor keys highlight the

correct item in each line and press ENTER .

Once the screen looks like this press CLEAR

Notice:There can only be one item in each line highlighted.


x

PART 1

1.Y1 = 3 cos (xº)

Y2 = 4 sin (xº)

Y3 = Y1(X) + Y2(X)

90 180 270 360

y

-90

MAXValue

x coordat MIN. Value

Y1 = 3 cos (xº)

Y2 = 4 sin (xº)

Y3 = Y1(x) + Y2(x)

Sketches of all three functions

How do I get Min / Max values from the TI-83 ?

See Calculator Hint Sheet 6

How do I define a function in terms of one already defined?See Calculator Hint Sheet 5

How do I get a table of values?See Calculator Hint Sheet 3

Suggested function forY1 + Y2.

In the form:-k cos(x - �� )

___ cos(x - ___)

3 cos (xº)

0º 30º 60º 90º 120º 150º 180º 210º 240º 270º 300º 330º 360ºxºTable of Values

4 sin (xº)


Y1(X) + Y2(X)

0º 30º 60º 90º 120º 150º 180º 210º 240º 270º 300º 330º 360ºxº

MIN.Value

x coordat MAX.

Value


2.

x

Y1 = 8 cos (xº)

Y2 = 6 sin (xº)

Y3 = Y1(X) + Y2(X)

90 180 270 360

y

-90

MAXValue


Y1 = 8 cos (xº)

Y2 = 6 sin (xº)

Y3 = Y1(x) + Y2(x)




___ cos(x - ___)

8 cos (xº)


6 sin (xº)


Y1(X) + Y2(X)


MIN.Value

x coordat MAX.

Value


x

Y1 = 5 cos (xº)

Y2 = 12 sin (xº)

Y3 = Y1(X) + Y2(X)

90 180 270 360

y

-90

MAXValue


Y1 = 5 cos (xº)

Y2 = 12 sin (xº)

Y3 = Y1(x) + Y2(x)




___ cos(x - ___)

5 cos (xº)


12 sin (xº)


Y1(X) + Y2(X)


MIN.Value

x coordat MAX.

Value

3.


x

Y1 = 12 cos (xº)

Y2 = 5 sin (xº)

Y3 = Y1(X) + Y2(X)

90 180 270 360

y

-90

MAXValue


Y1 = 12 cos (xº)

Y2 = 5 sin (xº)

Y3 = Y1(x) + Y2(x)




___ cos(x - ___)

12 cos (xº)


5 sin (xº)


Y1(X) + Y2(X)


MIN.Value

x coordat MAX.

Value

4.


a cos(x) + b sin(x) = k cos(x - ��α�) k tan�� α

3 cos(x) + 4 sin(x) = 5 cos(x - 53.13)

8 cos(x) + 6 sin(x) = 10cos(x - 36.90)

5 cos(x) + 12sin(x) = 13cos(x - 67.38)

12cos(x) + 5 sin(x) = 13cos(x - 22.62)

5.

Conjecture

a cos(x) + b sin(x) = k cos(x - ��α�)

k = _______________________ tan��α =__________


PART 2

y x x��

3 4cos sinSketch the graph of = + on your calculator

First enter the function as shown

Sketch the graph and pressuse number four to calculatethe maximum

Press to get

Enter a guessed value to the left of the maximumpress

Enter a guessed value to the right of the maximumpress

Simply press at the guess section to get

2nd

ENTER

CALC

trace

ENTER

ENTER

ENTER


90 180 270 360

y

-90

k xcos( )�

Use your graphic calculator to find the maximum and minimum y values

maximum = minimum =

Use your graphic calculator to find the coordinates of the maximum value

X coordinate = Y coordinate =

Suggest a function of the form that would give the values found above and the same graph

k � � �

On the axes below draw a graph of

−α

α

3 cos x + 4 sin x

T3 Scotland Wave Function of 12

90 180 270 360

y

-90

k xcos( )−α

Use your graphic calculator to find the maximum and minimum y values

maximum = minimum =

Use your graphic calculator to find the coordinates of the maximum value

X coordinate = Y coordinate =

Suggest a function of the form that would give the values found above and the same graph

k = α =

On the axes below draw a graph of8 6cos sinx x+


a cos(x) + b sin(x) = k cos(x - ��α�) k tanα��

3 cos(x) + 4 sin(x) = 5 cos(x - 53.13)

8 cos(x) + 6 sin(x) = 10cos(x - 36.90)

5 cos(x) + 12sin(x) = __cos(x - 67.38)

12cos(x) + 5 sin(x) = 13cos(x - _____)

Conjecture

a cos(x) + b sin(x) = k cos(x - ��α)

k = _______________________ tan��α =__________

Complete the table below

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