Graphs of Sine and Cosine Functions

Lesson 2.5

2

Ordered Pairs

Consider the values for x and y in the table to the right

Note Period = 2π Maximum y values Minimum y values

x sin(x) cos(x)

-3.1416 0.0000 -1.0000

-2.6180 -0.5000 -0.8660

-2.0944 -0.8660 -0.5000

-1.5708 -1.0000 0.0000

-1.0472 -0.8660 0.5000

-0.5236 -0.5000 0.8660

0.0000 0.0000 1.0000

0.5236 0.5000 0.8660

1.0472 0.8660 0.5000

1.5708 1.0000 0.0000

2.0944 0.8660 -0.5000

2.6180 0.5000 -0.8660

3.1416 0.0000 -1.0000

3.6652 -0.5000 -0.8660

4.1888 -0.8660 -0.5000

4.7124 -1.0000 0.0000

5.2360 -0.8660 0.5000

5.7596 -0.5000 0.8660

6.2832 0.0000 1.0000

3

Graphing the Ordered Pairs

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-6.28 -3.14 0.00 3.14 6.28 9.42

sin(x)

cos(x)

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-6.28 -3.14 0.00 3.14 6.28 9.42

sin(x)

cos(x)

Period = 2πPeriod = 2π

Maximum and minimum values

Maximum and minimum values

4

Graphing on Calculator

Go to ♦Y= screen Enter function

Choose F2, zoom 7-Trig

Graph is plotted Tic marks are in

units of π/2Try Web Graphing

Utility

5

Amplitude

Defined as the absolute value of maximum or minimum of the function

Try graphingy = 2 cos x What is the amplitude

For y = a cos x or y = a sin x The amplitude is |a|

amplitude = 1amplitude = 1

6

Period of a Trig Function(Recall slide from previous lesson)

The functions repeat themselves The period is the smallest value, p for

which f(x) = f(x + p)

For sin, cos, sec, csc The period is 2π

For tan and ctn The period is π

7

Period of a Trig Function

What happens for ? Try graphing y = sin 3x

What is the period?

Try y = cos 0.5x What is the period?

For

Period =

siny b x

siny b x

2

b

Same for cos, sec, cscSame for cos, sec, csc

8

Period of a Trig Function

For tangent Note amplitude

is without bound Period is π

For

Period =

tany xtany x

tany b x

b

• Predict the period fory = tan (1/3 x)

• Graph it and verify your prediction

• Predict the period fory = tan (1/3 x)

• Graph it and verify your prediction

9

Assignment

Lesson 2.5 Page 177 Exercises 1 – 61 EOO

also 63