Graphs of Sine and Cosine Functions

You’ll need graph paper

4.5

On Graph paper use a radius of 7in to represent the radius of your unit circle.

Then give both the fractional and decimal value of your trig function for each value of theta on the unit circle. (Do not include the last value of theta for your quadrant) i.e. 90, 180, 270, 360

2

Unit Circle Activity

Groups 1-4 – Sine Groups 5-8 – Cosine Groups 9-12 – Tangent

Group 4n+1 – Q1 Group 4n+2 – Q2 Group 4n+3 – Q3 Group 4n+4 – Q4

3

12 Groups

Ex: Group 7:7 = 4(1)+3

Cosine Q3

n is a whole #

Only need one graph

per group

Label with “as increases, [trigfunction( )] increases/decreases.”

Ex for sine of Q1: “as increases, increases.”

4

sin

5

Graph the sine and cosine functions Graph the sine function on a new piece of graph

paper

Label your x-axis in radians in multiples of . Use 1 square for each measure

On your y-axis label . Count 1 square as ¼.

Then graph the cosine function a separate

axis. (use the same labels)

6

2,2

3,1,

2

1,0

2

6to

Everyone needs to

do their own!

Everyone is starting

with sine

13 squares on each side of y-axis

8 squares on each side of x-axis

Graph the csc and sec functions

Graph csc with the sin graph and sec with the cos graph

Then make a new graph for tan and cot Label your x-axis in radians in multiples of

7

12

6

5

4

3

3

2

2346

8

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

9

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

10

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

11

Amplitude

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

Try graphingy = 2 sinx What is the amplitude

For y = a cos x or y = a sin x The amplitude is |a| Do we need to worry about the amplitude for the other

trig functinos?

amplitude = 1amplitude = 1 y=sinxy=sinx

2

12

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 cot The period is π

13

Period of a Trig Function

What happens for ? Try graphing y = sin 2x

What is the period? What about y = sin 3x

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

2

2

3

2

4

2/1

2

14

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 for

y = tan (1/3 x)

• Graph it and verify your prediction

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

• Graph it and verify your prediction

3

3/1

Same for cotSame for cot

Review of Transformations

15

1)82(3)( 2 xxf

1)]4(2[3)( 2 xxf

Or reflection over x-axis!

Or reflection over y-axis!

Review of Transformations

16

Sketch the graphDo non-rigid transformation 1st (strech/compress)Then rigid transformations (up/down and left/right)

17

Let’s investigate with graphs of trig functions!

18

dcxbay )](sin[

Desmos.com

Practice time

Graph

a.

b.

19

12cos2 xy

2sin3

xy

Sketch each transformation of the graph

Sketch between 0 and 2pi while doing transformations

Does the make a difference?

20

H Dub

4-5 Pg 328 #1-25odd, 35-51EOO

Graph the sine and cosine functions

Regraph the sine and cosine functions on two separate axis

Label your graph in radians

On your y-axis labelLeave room above and below!

On your x-axis label multiples of

21

1,2

1,0

360,270,225,180,135,90,45,0

2

6to