Aim: Graphs of y = sin x and y = cos x Course: Alg. 2 & Trig. Aim: What do the graphs of Trig...

Aim: Graphs of y = sin x and y = cos x Course: Alg. 2 & Trig.

Aim: What do the graphs of Trig functionslook like?

Do Now:You and a friend are the last people seated onFerris wheel. Once the ride begins, the wheelmoves at a constant speed.It takes 36 seconds to complete one revolution.

5’5’

40’

When the ride starts, howhigh above the ground are you?

At what height are you after 9 s.? after 18 s.? 27 s.?

At what height are you after 126 s.? How many revolutions have you made?Predict where you will be after 3 minutes.


Periodic Functions

What would a graph showing the relationshipbetween your height above the ground and the time since the ride began. Use 0 ≤ t ≤ 144for the domain, where t = 0 is the time whenthe ride began.

5

10

15

20

25

30

35

40

0 24 48 72 96 120 144


Periodic Functions

5

10

15

20

25

30

35

40

• A periodic function repeats a pattern of y-values (outputs) at regular intervals.

• A period of a function is the horizontal length of one cycle.

0 24 48 72 96 120 144

• One complete pattern is called a cycle,which may begin anywhere on the graph.

1 cycle 1 cycle

36 s. 36 s.


Periodic Functions

5

10

15

20

25

30

35

40

0 24 48 72 96 120 144

The amplitude of a periodic function is halfthe difference between the minimum andmaximum values of the function.

max 45’

min 5’

amplitude

y = 20


y = sin x

unit circle y = sin x1

-1

π 2π3π/2 π/2

For what value of x does the graph y = sinx reach the maximum amplitude?

sine curve or wave

What is the cycle?

What is the period?

π/2

360º or 2π

360º or 2π

radians


Model Problem

y = sin x1

-1

π 2π3π/2 π/2

Determine whether sin x increases or decrease in each quadrant.

sine curve or wave

radians

QI – increasing from 0 to 1QII – decreasing from 1 to 0QIII – decreasing from 0 to -1QIV – increasing from -1 to 0

QI QII QIII QIV


y = cos x

unit circle y = cos x1

-1

π 2π3π/2 π/2

For what value of x does the graph y = cos x reach the maximum amplitude?

cosine curve

What is the cycle?

What is the period?

0 & 2π

360º or 2π

360º or 2π

radians

-1


Model Problem

y = cos x1

-1

π 2π3π/2 π/2

Determine whether cos x increases or decrease in each quadrant.

cosine curve

radians

QI – decreasing from 1 to 0QII – decreasing from 0 to -1QIII – increasing from -1 to 0QIV – increasing from 0 to 1

QI QII QIII QIV


Comparing sine and cosine curves

y = sin x1

-1

π 2π3π/2 π/2

radians

y = cos x1

-1

π 2π3π/2 π/2radians

Both curves have amplitudes of 1 and maximums of 1 and minimums of -1.


Comparing sine and cosine curves

y = sin x radians

y = cos x1

-1

π 2π 3π/2 π/2radians

1

-1

π 2π 3π/2 π/2 π 2π 3π/2 π/2 π 3π/2 π/2

Both curves are cyclical and have periods of 2π.

period - 2π

period - 2π

cos x = cos(x + 2πk) for any integer k

sin x = sin(x + 2πk) for any integer k


Comparing sine and cosine curvesy = sin x1

-1

π 2π3π/2 π/2

radians

y = cos x1

-1


The cosine curve is a translation of the sine curve

y cosxT

2,0

y sinx

y cos(x 2

) sinx

sin = cos(90º – ) cos = sin(90º – )Co-


Sine Curve & Trig Values

y = sin x1

-1

π 2π3π/2 π/2

radians

x 0

0 30 60 90 120

150 0

210

240

270

300

330

360

y 0 1 0 -1 0

6

3

2

2

3

5

6

7

6

4

3

3

2

5

3

11

6

2

1

23

2

3

23

2

3

2

1

2

1

2

1

2

Domain = | Real numbers

Range = 1 1

x x

y


Cosine Curve & Trig Values

x 0

0 30 60 90 120

150 0

210

240

270

300

330

360

y 1 0 -1 1 1

6

3

2

2

3

5

6

7

6

4

3

3

2

5

3

11

6

2

1

23

2

3

23

2

3

2

1

2

1

2

1

2

y = cos x1

-1


Domain = | Real numbers

Range = 1 1

x x

y

Aim: Graphs of y = sin x and y = cos x Course: Alg. 2 & Trig. Aim: What do the graphs of Trig...

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Transcript of Aim: Graphs of y = sin x and y = cos x Course: Alg. 2 & Trig. Aim: What do the graphs of Trig...