Graphing the sine and cosine functions.

Graphing the sine and cosine functions. 1 / 9

Today’s goal:

Our goal today is to graph the sine and cosine functions and functionsrelated to them.

Here is our plan of attack. Since sin(x) is periodic with period 2π we willgraph sin(x) by (1)

graphing it over [0, 2π]

and then (2)

repeating every2π.

After that we will do the same for cosine. Finally we will study functionsof the form A · sin(ωx) and A · cos(ωx)

Graphing the sine and cosine functions. 2 / 9

Today’s goal:

Our goal today is to graph the sine and cosine functions and functionsrelated to them.Here is our plan of attack. Since sin(x) is periodic with period 2π we willgraph sin(x) by (1)

graphing it over [0, 2π]

and then (2)

repeating every2π.

After that we will do the same for cosine. Finally we will study functionsof the form A · sin(ωx) and A · cos(ωx)

Graphing the sine and cosine functions. 2 / 9

Today’s goal:

Our goal today is to graph the sine and cosine functions and functionsrelated to them.Here is our plan of attack. Since sin(x) is periodic with period 2π we willgraph sin(x) by (1) graphing it over [0, 2π] and then (2)

repeating every2π.

After that we will do the same for cosine. Finally we will study functionsof the form A · sin(ωx) and A · cos(ωx)

Graphing the sine and cosine functions. 2 / 9

Today’s goal:

Our goal today is to graph the sine and cosine functions and functionsrelated to them.Here is our plan of attack. Since sin(x) is periodic with period 2π we willgraph sin(x) by (1) graphing it over [0, 2π] and then (2)

repeating every2π.

After that we will do the same for cosine. Finally we will study functionsof the form A · sin(ωx) and A · cos(ωx)

Graphing the sine and cosine functions. 2 / 9

Today’s goal:

Our goal today is to graph the sine and cosine functions and functionsrelated to them.Here is our plan of attack. Since sin(x) is periodic with period 2π we willgraph sin(x) by (1) graphing it over [0, 2π] and then (2)

repeating every2π.

After that we will do the same for cosine. Finally we will study functionsof the form A · sin(ωx) and A · cos(ωx)

Graphing the sine and cosine functions. 2 / 9

Today’s goal:

Our goal today is to graph the sine and cosine functions and functionsrelated to them.Here is our plan of attack. Since sin(x) is periodic with period 2π we willgraph sin(x) by (1) graphing it over [0, 2π] and then (2) repeating every2π.

After that we will do the same for cosine. Finally we will study functionsof the form A · sin(ωx) and A · cos(ωx)

Graphing the sine and cosine functions. 2 / 9

Today’s goal:

Our goal today is to graph the sine and cosine functions and functionsrelated to them.Here is our plan of attack. Since sin(x) is periodic with period 2π we willgraph sin(x) by (1) graphing it over [0, 2π] and then (2) repeating every2π.

After that we will do the same for cosine. Finally we will study functionsof the form A · sin(ωx) and A · cos(ωx)

Graphing the sine and cosine functions. 2 / 9

Today’s goal:

Our goal today is to graph the sine and cosine functions and functionsrelated to them.Here is our plan of attack. Since sin(x) is periodic with period 2π we willgraph sin(x) by (1) graphing it over [0, 2π] and then (2) repeating every2π.

After that we will do the same for cosine. Finally we will study functionsof the form A · sin(ωx) and A · cos(ωx)

Graphing the sine and cosine functions. 2 / 9

Graphing sin(x)

Lets try to graph y = sin(x) over the interval [0, 2π].Start by filling in this table:

x 0 π/6 π/4 π/3 π/2 2π/3 3π/4 5π/6 π

sin(x)exact

0 12

√22

√32 1

√32

√22

12 0

decimal.

0 .5 .7 .86 1 .86 .7 .5 0

plot these points on the graph below. The dashed in the x-direction are atπ/6, π/4, . . . . In the y -direction they are every .2

π 2π

Graphing the sine and cosine functions. 3 / 9

Graphing sin(x)

Lets try to graph y = sin(x) over the interval [0, 2π].Start by filling in this table:

x 0 π/6 π/4 π/3 π/2 2π/3 3π/4 5π/6 π

sin(x)exact

0

12

√22

√32 1

√32

√22

12 0

decimal.

0 .5 .7 .86 1 .86 .7 .5 0

plot these points on the graph below. The dashed in the x-direction are atπ/6, π/4, . . . . In the y -direction they are every .2

π 2π

Graphing the sine and cosine functions. 3 / 9

Graphing sin(x)

Lets try to graph y = sin(x) over the interval [0, 2π].Start by filling in this table:

x 0 π/6 π/4 π/3 π/2 2π/3 3π/4 5π/6 π

sin(x)exact

0 12

√22

√32 1

√32

√22

12 0

decimal.

0 .5 .7 .86 1 .86 .7 .5 0

plot these points on the graph below. The dashed in the x-direction are atπ/6, π/4, . . . . In the y -direction they are every .2

π 2π

Graphing the sine and cosine functions. 3 / 9

Graphing sin(x)

Lets try to graph y = sin(x) over the interval [0, 2π].Start by filling in this table:

x 0 π/6 π/4 π/3 π/2 2π/3 3π/4 5π/6 π

sin(x)exact

0 12

√22

√32 1

√32

√22

12 0

decimal.

0 .5 .7 .86 1 .86 .7 .5 0

plot these points on the graph below. The dashed in the x-direction are atπ/6, π/4, . . . . In the y -direction they are every .2

π 2π

Graphing the sine and cosine functions. 3 / 9

Graphing sin(x)

Lets try to graph y = sin(x) over the interval [0, 2π].Start by filling in this table:

x 0 π/6 π/4 π/3 π/2 2π/3 3π/4 5π/6 π

sin(x)exact

0 12

√22

√32

1√32

√22

12 0

decimal.

0 .5 .7 .86 1 .86 .7 .5 0

plot these points on the graph below. The dashed in the x-direction are atπ/6, π/4, . . . . In the y -direction they are every .2

π 2π

Graphing the sine and cosine functions. 3 / 9

Graphing sin(x)

Lets try to graph y = sin(x) over the interval [0, 2π].Start by filling in this table:

x 0 π/6 π/4 π/3 π/2 2π/3 3π/4 5π/6 π

sin(x)exact

0 12

√22

√32 1

√32

√22

12 0

decimal.

0 .5 .7 .86 1 .86 .7 .5 0

plot these points on the graph below. The dashed in the x-direction are atπ/6, π/4, . . . . In the y -direction they are every .2

π 2π

Graphing the sine and cosine functions. 3 / 9

Graphing sin(x)

Lets try to graph y = sin(x) over the interval [0, 2π].Start by filling in this table:

x 0 π/6 π/4 π/3 π/2 2π/3 3π/4 5π/6 π

sin(x)exact

0 12

√22

√32 1

√32

√22

12 0

decimal.

0 .5 .7 .86 1 .86 .7 .5 0

plot these points on the graph below. The dashed in the x-direction are atπ/6, π/4, . . . . In the y -direction they are every .2

π 2π

Graphing the sine and cosine functions. 3 / 9

Graphing sin(x)

Lets try to graph y = sin(x) over the interval [0, 2π].Start by filling in this table:

x 0 π/6 π/4 π/3 π/2 2π/3 3π/4 5π/6 π

sin(x)exact

0 12

√22

√32 1

√32

√22

12 0

decimal.

0 .5 .7 .86 1 .86 .7 .5 0

plot these points on the graph below. The dashed in the x-direction are atπ/6, π/4, . . . . In the y -direction they are every .2

π 2π

Graphing the sine and cosine functions. 3 / 9

Graphing sin(x)

Lets try to graph y = sin(x) over the interval [0, 2π].Start by filling in this table:

x 0 π/6 π/4 π/3 π/2 2π/3 3π/4 5π/6 π

sin(x)exact

0 12

√22

√32 1

√32

√22

12

0

decimal.

0 .5 .7 .86 1 .86 .7 .5 0

plot these points on the graph below. The dashed in the x-direction are atπ/6, π/4, . . . . In the y -direction they are every .2

π 2π

Graphing the sine and cosine functions. 3 / 9

Graphing sin(x)

Lets try to graph y = sin(x) over the interval [0, 2π].Start by filling in this table:

x 0 π/6 π/4 π/3 π/2 2π/3 3π/4 5π/6 π

sin(x)exact

0 12

√22

√32 1

√32

√22

12 0

decimal.

0 .5 .7 .86 1 .86 .7 .5 0

plot these points on the graph below. The dashed in the x-direction are atπ/6, π/4, . . . . In the y -direction they are every .2

π 2π

Graphing the sine and cosine functions. 3 / 9

Graphing sin(x)

Lets try to graph y = sin(x) over the interval [0, 2π].Start by filling in this table:

x 0 π/6 π/4 π/3 π/2 2π/3 3π/4 5π/6 π

sin(x)exact

0 12

√22

√32 1

√32

√22

12 0

decimal. 0 .5 .7 .86 1 .86 .7 .5 0plot these points on the graph below. The dashed in the x-direction are atπ/6, π/4, . . . . In the y -direction they are every .2

π 2π

Graphing the sine and cosine functions. 3 / 9

Graphing sin(x)

Lets try to graph y = sin(x) over the interval [0, 2π].Start by filling in this table:

x 0 π/6 π/4 π/3 π/2 2π/3 3π/4 5π/6 π

sin(x)exact

0 12

√22

√32 1

√32

√22

12 0

decimal. 0 .5 .7 .86 1 .86 .7 .5 0plot these points on the graph below. The dashed in the x-direction are atπ/6, π/4, . . . . In the y -direction they are every .2

π 2π

Graphing the sine and cosine functions. 3 / 9

Graphing sin(x)

Lets try to graph y = sin(x) over the interval [0, 2π].Start by filling in this table:

x 0 π/6 π/4 π/3 π/2 2π/3 3π/4 5π/6 π

sin(x)exact

0 12

√22

√32 1

√32

√22

12 0

decimal. 0 .5 .7 .86 1 .86 .7 .5 0plot these points on the graph below. The dashed in the x-direction are atπ/6, π/4, . . . . In the y -direction they are every .2

π 2π

Graphing the sine and cosine functions. 3 / 9

Graphing sin(x)

Lets try to graph y = sin(x) over the interval [0, 2π].Start by filling in this table:

x 0 π/6 π/4 π/3 π/2 2π/3 3π/4 5π/6 π

sin(x)exact

0 12

√22

√32 1

√32

√22

12 0

decimal. 0 .5 .7 .86 1 .86 .7 .5 0plot these points on the graph below. The dashed in the x-direction are atπ/6, π/4, . . . . In the y -direction they are every .2

π 2π

Graphing the sine and cosine functions. 3 / 9

Graphing sin(x)

Lets try to graph y = sin(x) over the interval [0, 2π].Start by filling in this table:

x 0 π/6 π/4 π/3 π/2 2π/3 3π/4 5π/6 π

sin(x)exact

0 12

√22

√32 1

√32

√22

12 0

decimal. 0 .5 .7 .86 1 .86 .7 .5 0plot these points on the graph below. The dashed in the x-direction are atπ/6, π/4, . . . . In the y -direction they are every .2

π 2π

Graphing the sine and cosine functions. 3 / 9

Graphing sine

Here is the graph of sin(x) so far:

π 2π

We have the identity: − sin(x − π) = sin(x).

We can graph − sin(x − π) by shifting right by π and then flipping overthe x-axis.We can now get the whole graph by “connecting the dots”

Graphing the sine and cosine functions. 4 / 9

Graphing sine

Here is the graph of sin(x) so far:

π 2π

We have the identity: − sin(x − π) = sin(x).

We can graph − sin(x − π) by shifting right by π and then flipping overthe x-axis.We can now get the whole graph by “connecting the dots”

Graphing the sine and cosine functions. 4 / 9

Graphing sine

Here is the graph of sin(x) so far:

π 2π

We have the identity: − sin(x − π) = sin(x).We can graph − sin(x − π) by shifting right by π and then flipping overthe x-axis.

We can now get the whole graph by “connecting the dots”

Graphing the sine and cosine functions. 4 / 9

Graphing sine

Here is the graph of sin(x) so far:

π 2π

We have the identity: − sin(x − π) = sin(x).We can graph − sin(x − π) by shifting right by π and then flipping overthe x-axis.

We can now get the whole graph by “connecting the dots”

Graphing the sine and cosine functions. 4 / 9

Graphing sine

Here is the graph of sin(x) so far:

π 2π

We have the identity: − sin(x − π) = sin(x).We can graph − sin(x − π) by shifting right by π and then flipping overthe x-axis.

We can now get the whole graph by “connecting the dots”

Graphing the sine and cosine functions. 4 / 9

Graphing sine

Here is the graph of sin(x) so far:

π 2π

We have the identity: − sin(x − π) = sin(x).We can graph − sin(x − π) by shifting right by π and then flipping overthe x-axis.

We can now get the whole graph by “connecting the dots”

Graphing the sine and cosine functions. 4 / 9

Graphing sine

Here is the graph of sin(x) so far:

π 2π

We have the identity: − sin(x − π) = sin(x).We can graph − sin(x − π) by shifting right by π and then flipping overthe x-axis.We can now get the whole graph by “connecting the dots”

Graphing the sine and cosine functions. 4 / 9

Graphing sine

Here is the graph of sin(x) so far:

π 2π

We have the identity: − sin(x − π) = sin(x).We can graph − sin(x − π) by shifting right by π and then flipping overthe x-axis.We can now get the whole graph by “connecting the dots”

Graphing the sine and cosine functions. 4 / 9

Finishing the graph of sine.Here is the graph of sin(x) over the interval [0, 2π].

2π−2π π 3π 4π−1π

Now we take advantage of periodicity: In order to graph sin(x) we justneed to draw the same “sine-wave” every 2π.

Now we are ready to use this graph to make some conclusions:

1 What is the maximum value? What input(s)? achieve it?2 What is the minimum value? What input(s)? achieve it?3 Even or odd?4 Where in sin(x) increasing?5 Where in sin(x) decreasing?6 Where are the turning points?

Graphing the sine and cosine functions. 5 / 9

Finishing the graph of sine.Here is the graph of sin(x) over the interval [0, 2π].

2π−2π π 3π 4π−1π

Now we take advantage of periodicity: In order to graph sin(x) we justneed to draw the same “sine-wave” every 2π.Now we are ready to use this graph to make some conclusions:

1 What is the maximum value? What input(s)? achieve it?2 What is the minimum value? What input(s)? achieve it?3 Even or odd?4 Where in sin(x) increasing?5 Where in sin(x) decreasing?6 Where are the turning points?

Graphing the sine and cosine functions. 5 / 9

Finishing the graph of sine.Here is the graph of sin(x) over the interval [0, 2π].

2π−2π π 3π 4π−1π

Now we take advantage of periodicity: In order to graph sin(x) we justneed to draw the same “sine-wave” every 2π.Now we are ready to use this graph to make some conclusions:

1 What is the maximum value? What input(s)? achieve it?2 What is the minimum value? What input(s)? achieve it?3 Even or odd?4 Where in sin(x) increasing?5 Where in sin(x) decreasing?6 Where are the turning points?

Graphing the sine and cosine functions. 5 / 9

Getting Cosine from the graph of sine.

Now, we could graph cosine by doing the same work: plotting somepoints, using symmetries, connecting the dots.

Instead let’s just use the graph of sine and some geometryWhat is sin

(x + π

2

)? Look at your identities sheet.

sin(x + π

2

)=

cos(x)

.

In order to graph cos(x) = sin(x + π

2

), all we need to do is start with the

graph of sin and shift π2 units to the left.

2π−2π π 3π 4π−1π

Graphing the sine and cosine functions. 6 / 9

Getting Cosine from the graph of sine.

Now, we could graph cosine by doing the same work: plotting somepoints, using symmetries, connecting the dots.Instead let’s just use the graph of sine and some geometry

What is sin(x + π

2

)? Look at your identities sheet.

sin(x + π

2

)=

cos(x)

.

In order to graph cos(x) = sin(x + π

2

), all we need to do is start with the

graph of sin and shift π2 units to the left.

2π−2π π 3π 4π−1π

Graphing the sine and cosine functions. 6 / 9

Getting Cosine from the graph of sine.

Now, we could graph cosine by doing the same work: plotting somepoints, using symmetries, connecting the dots.Instead let’s just use the graph of sine and some geometryWhat is sin

(x + π

2

)? Look at your identities sheet.

sin(x + π

2

)=

cos(x)

.

In order to graph cos(x) = sin(x + π

2

), all we need to do is start with the

graph of sin and shift π2 units to the left.

2π−2π π 3π 4π−1π

Graphing the sine and cosine functions. 6 / 9

Getting Cosine from the graph of sine.

Now, we could graph cosine by doing the same work: plotting somepoints, using symmetries, connecting the dots.Instead let’s just use the graph of sine and some geometryWhat is sin

(x + π

2

)? Look at your identities sheet.

sin(x + π

2

)= cos(x).

In order to graph cos(x) = sin(x + π

2

), all we need to do is start with the

graph of sin and shift π2 units to the left.

2π−2π π 3π 4π−1π

Graphing the sine and cosine functions. 6 / 9

Getting Cosine from the graph of sine.

Now, we could graph cosine by doing the same work: plotting somepoints, using symmetries, connecting the dots.Instead let’s just use the graph of sine and some geometryWhat is sin

(x + π

2

)? Look at your identities sheet.

sin(x + π

2

)= cos(x).

In order to graph cos(x) = sin(x + π

2

), all we need to do is start with the

graph of sin and shift π2 units to the left.

2π−2π π 3π 4π−1π

Graphing the sine and cosine functions. 6 / 9

Getting Cosine from the graph of sine.

Now, we could graph cosine by doing the same work: plotting somepoints, using symmetries, connecting the dots.Instead let’s just use the graph of sine and some geometryWhat is sin

(x + π

2

)? Look at your identities sheet.

sin(x + π

2

)= cos(x).

In order to graph cos(x) = sin(x + π

2

), all we need to do is start with the

graph of sin and shift π2 units to the left.

2π−2π π 3π 4π−1π

Graphing the sine and cosine functions. 6 / 9

Properties of cosine?

2π−2π π 3π 4π−1π

1 What is the maximum value? What input(s)? achieve it?

2 What is the minimum value? What input(s)? achieve it?

3 Even or odd?

4 Where in cos(x) increasing?

5 Where in cos(x) decreasing?

6 Where are the turning points?

Graphing the sine and cosine functions. 7 / 9

Scaling

Here are cosine and sine:

How can we get the graph of 2 · cos(x ·3)?

Start with cosine. Stretch vertically by a factor of 2. Compresshorizontally by a factor of 3.

For you: Plot 3 · sin(2x)

Graphing the sine and cosine functions. 8 / 9

Scaling

Here are cosine and sine:

How can we get the graph of 2 · cos(x ·3)?Start with cosine.

Stretch vertically by a factor of 2. Compresshorizontally by a factor of 3.

For you: Plot 3 · sin(2x)

Graphing the sine and cosine functions. 8 / 9

Scaling

Here are cosine and sine:

How can we get the graph of 2 · cos(x ·3)?Start with cosine.

Stretch vertically by a factor of 2. Compresshorizontally by a factor of 3.

For you: Plot 3 · sin(2x)

Graphing the sine and cosine functions. 8 / 9

Scaling

Here are cosine and sine:

How can we get the graph of 2 · cos(x ·3)?Start with cosine. Stretch vertically by a factor of 2.

Compresshorizontally by a factor of 3.

For you: Plot 3 · sin(2x)

Graphing the sine and cosine functions. 8 / 9

Scaling

Here are cosine and sine:

How can we get the graph of 2 · cos(x ·3)?Start with cosine. Stretch vertically by a factor of 2.

Compresshorizontally by a factor of 3.

For you: Plot 3 · sin(2x)

Graphing the sine and cosine functions. 8 / 9

Scaling

Here are cosine and sine:

How can we get the graph of 2 · cos(x ·3)?Start with cosine. Stretch vertically by a factor of 2. Compresshorizontally by a factor of 3.

For you: Plot 3 · sin(2x)

Graphing the sine and cosine functions. 8 / 9

Scaling

Here are cosine and sine:

How can we get the graph of 2 · cos(x ·3)?Start with cosine. Stretch vertically by a factor of 2. Compresshorizontally by a factor of 3.

For you: Plot 3 · sin(2x)

Graphing the sine and cosine functions. 8 / 9

Scaling

Here are cosine and sine:

How can we get the graph of 2 · cos(x ·3)?Start with cosine. Stretch vertically by a factor of 2. Compresshorizontally by a factor of 3.

For you: Plot 3 · sin(2x)

Graphing the sine and cosine functions. 8 / 9

The period of A · cos(ωx)

How long does it take for cos(2 · x) to repeat?

(π, 0) (2π, 0)(−π, 0)(−2π, 0)

Repeats every π. The period is half of what is used to be.

Theorem

The period of cos(ωx) and of sin(ωx) is 2π/ω.

Sketch a graph of sin(πx). What is its period?

Graphing the sine and cosine functions. 9 / 9

The period of A · cos(ωx)

How long does it take for cos(2 · x) to repeat?

(π, 0) (2π, 0)(−π, 0)(−2π, 0)

Repeats every π. The period is half of what is used to be.

Theorem

The period of cos(ωx) and of sin(ωx) is 2π/ω.

Sketch a graph of sin(πx). What is its period?

Graphing the sine and cosine functions. 9 / 9

The period of A · cos(ωx)

How long does it take for cos(2 · x) to repeat?

(π, 0) (2π, 0)(−π, 0)(−2π, 0)

Repeats every π. The period is half of what is used to be.

Theorem

The period of cos(ωx) and of sin(ωx) is 2π/ω.

Sketch a graph of sin(πx). What is its period?

Graphing the sine and cosine functions. 9 / 9

The period of A · cos(ωx)

How long does it take for cos(2 · x) to repeat?

(π, 0) (2π, 0)(−π, 0)(−2π, 0)

Repeats every π.

The period is half of what is used to be.

Theorem

The period of cos(ωx) and of sin(ωx) is 2π/ω.

Sketch a graph of sin(πx). What is its period?

Graphing the sine and cosine functions. 9 / 9

The period of A · cos(ωx)

How long does it take for cos(2 · x) to repeat?

(π, 0) (2π, 0)(−π, 0)(−2π, 0)

Repeats every π. The period is half of what is used to be.

Theorem

The period of cos(ωx) and of sin(ωx) is 2π/ω.

Sketch a graph of sin(πx). What is its period?

Graphing the sine and cosine functions. 9 / 9

The period of A · cos(ωx)

How long does it take for cos(2 · x) to repeat?

(π, 0) (2π, 0)(−π, 0)(−2π, 0)

Repeats every π. The period is half of what is used to be.

Theorem

The period of cos(ωx) and of sin(ωx) is 2π/ω.

Sketch a graph of sin(πx). What is its period?

Graphing the sine and cosine functions. 9 / 9

The period of A · cos(ωx)

How long does it take for cos(2 · x) to repeat?

(π, 0) (2π, 0)(−π, 0)(−2π, 0)

Repeats every π. The period is half of what is used to be.

Theorem

The period of cos(ωx) and of sin(ωx) is 2π/ω.

Sketch a graph of sin(πx). What is its period?

Graphing the sine and cosine functions. 9 / 9