GRAPHS OF SINE AND COSINE FUNCTIONSObjectives:

Sketch the graphs of basic sine and cosine functions

Use amplitude, period and translations to help sketch the graphs of sine and cosine functions

GRAPHING SINE AND COSINE FUNCTIONS

In this unit, we will take the unit circle introduced in the previous unit, and graph it on the Cartesian plane.

To do this, we are going to “unravel” the unit circle. Recall that for the unit circle the coordinates are where is the central angle. To graph, rewrite the coordinates as where is the central angle, in radians. We expanded the sine coordinates for .

cos ,sin siny x

,sinx x x

3

4

SINE CURVE

SINE CURVE Notice that the curve ranges from 1 to -1.

The maximum value is 1, which is at . The minimum value is -1 at .

This “height” of the sine function is called the amplitude.

If we had continued the curve, it would repeat. This means that the sine curve is periodic. Look back at the unit circle, the sine value changes until it reaches . Therefore, the curve will repeat every units, making the period . The domain is all real numbers.

2x

3

2x

22 2

COSINE CURVE Similarly, when we expand the cosine curve,

from the unit circle, we have:

Notice that the range is also and the domain is also . The period is also

cosy x

1,1 , 2

BASIC SINE AND COSINE CURVESo Comparing , we see

that the curves are almost identical, except that the sine curve starts at and the cosine curve starts at .

and si cosn y xy x

0y 1y

CHARACTERISTICS OF SINE AND COSINE

One Cycle: the period of each of these curves is from . The curves repeat indefinitely to the left and right.

Domain: Range:

Symmetry: Sine curve is symmetric with respect to the origin (odd function) and Cosine curve is symmetric with respect to the y-axis (even function)

0,2

, 1,1

CHARACTERISTICS OF SINE AND COSINE

To sketch the graphs of the basic sine and cosine functions by hand, it helps to note five key points in one period of each graph: the intercepts, maximum points and minimum points.

Sine key points:

Cosine key points:

30,0 ; ,1 ; ,0 ; , 1 ; 2 ,0

2 2

30,1 ; ,0 ; , 1 ; ,0 ; 2 ,1

2 2

TRANSFORMATIONS OF SINE AND COSINE

You will study the graphic effect of each of the constants in equations of the forms:

Sine: Cosine:

Amplitude represents half the distance between the max and the min values of the function (multiply the y-values) If , the basic curve is stretched and if is

shrunk

Period the horizontal distance needed to complete one cycle of the

function If , the period of the graph is greater than

and if is less than

, , and a b c d

siny d a b x c cosy d a b x c

a

2

b

1a 0 1a

0 1b 21b 2

TRANSFORMATIONS OF SINE AND COSINE Phase shift: the c amount shifted right or

left to determine the new endpoints of the one cycle If , the graph has a horizontal translation to

the right and if to the left.

Vertical translation: the d amount shifted up or down to determine the new max and min of the one cycle If , the graph has a vertical translation up

and if the graph goes down

Reflection: the sign in front of the determines if it is reflected over the x-axis

a

0c 0c

0d 0d

EX: FIND THE AMPLITUDE, THE PERIOD, THE REFLECTION, ANY VERTICAL TRANSLATION, AND ANY PHASE SHIFT OF EACH GRAPH.

1.

2.

3.

sin 44

y x

2 2cosy x

1 13 sin

2 4y x

EX: SKETCH THE GRAPH OF ONE CYCLE 4. 3cos2y x

EX: SKETCH THE GRAPH OF ONE CYCLE

5. 4sin2

y x

EX: SKETCH THE GRAPH OF ONE CYCLE 6.

13 cos

2y x

EX: SKETCH THE GRAPH OF ONE CYCLE 7. 3 2sin 4

4y x