Graphing Sine and Cosine

Pre Calculus

What you will learnHow to graph sine and cosine functions. How to translate sine and cosine functions

(shift, left, right, vertical stretch, horizontal stretch)

How to use key points to “sketch” a graph.

PlanDiscuss how to use the Unit Circle to help

with graphingGraphing Sine and Cosine and their

translations

Fundamental Trigonometric Identities

Cofunction Identitiessin = cos(90 ) cos = sin(90 )sin = cos (π/2 ) cos = sin (π/2 )tan = cot(90 ) cot = tan(90 )tan = cot (π/2 ) cot = tan (π/2 )sec = csc(90 ) csc = sec(90 ) sec = csc (π/2 ) csc = sec (π/2 )

Reciprocal Identities

sin = 1/csc cos = 1/sec tan = 1/cot cot = 1/tan sec = 1/cos csc = 1/sin

Quotient Identities

tan = sin /cos cot = cos /sin

Pythagorean Identities

sin2 + cos2 = 1 tan2 + 1 = sec2 cot2 + 1 = csc2

Review of Even and Odd Functions

Cosine and secant functions are evencos (-t) = cos t sec (-t) = sec t

Sine, cosecant, tangent and cotangent are oddsin (-t) = -sin t csc (-t) = -csc ttan (-t) = -tan t cot (-t) = -cot t

Graphing – Sine and Cosine

Key Things to DiscussShape of the functions

Using the Unit Circle to help identify key pointsPeriodic Nature

Translations that are the same as other functions we have studied

Translations that are different than others we have studied

Using the calculator and correct interpretation of the calculator

Shape of Sine and CosineThe unit circle: we imagined the real

number line wrapped around the circle.Each real number corresponded to a point

(x, y) which we found to be the (cosine, sine) of the angle represented by the real number.

To graph the sine and cosine we can go back to the unit circle to find the ordered pairs for our graph.

Let’s convert some of these numbers to decimal form – start with cosine

87.2

3

11

5.2

1

71.2

2

87.2

3

11

00

00

87.2

3 87.

2

3

71.2

2

71.2

2

71.2

2

5.2

1

5.2

1

5.2

1

Cosine Key Features to define the shape

Input: x

Angle

Output:

Cos x

0

1

max

π/6

.87

π/4

.71

π/3

.5

π/2

0

int

2π/3

-.5

3π/4

-.71

5π/6

-.87

Input: x

Angle

Output:

Cos x

π

-1

min

7π/6

-.87

5π/4

-.71

4π/3

-.5

3π/2

0

int

5π/3

.5

7π/4

.71

11π/6

.87

2π

1

max

ShapeCalculatorMode: RadiansWindow:

Xmin 0 Xmax 2Xscl /2Ymin -2Ymax +2Yscl .5

Y=cos x

Cosine

For the function: cosyThe angle is the input or independent variable and the cosine

ratio is the output or dependent variable.

From a unit circle perspective, the input is the angle and the output

is the “x” coordinate of the ordered pair.

Remember “coterminal angles” every 2π the values will repeat

– this is called a periodic function

We will use the interval [0, 2π] as the reference period.

Graph of the Cosine Function

To sketch the graph of y = cos x first locate the key points.These are the maximum points, the minimum points, and the intercepts.

10-101cos x

0x2

2

32

Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period.

y

2

3

2

22

32

2

5

1

1

x

y = cos x

WorksheetLet’s graph it!

Now let’s look at Sine

Let’s convert some of these numbers to decimal form – start with sine

87.2

311

5.2

1

71.2

2

87.2

3

11

00 00

87.2

3 87.

2

3

71.2

2

71.2

2

71.2

2

5.2

1 5.

2

1

5.2

1

Sine Key Features to define the shape

Input: x

Angle

Output

sin x

0

0

int

π/6

.5

π/4

.71

π/3

.87

π/2

1

max

2π/3

.87

3π/4

.71

5π/6

.5

Input: x

Angle

Output:

sin x

π

0

int

7π/6

-.5

5π/4

-.71

4π/3

-.87

3π/2

-1

min

5π/3

-.87

7π/4

-.71

11π/6

-.5

2π

0

int

ShapeCalculatorMode: RadiansWindow:

Xmin 0 Xmax 2Xscl /2Ymin -2Ymax +2Yscl .5

Y=sin x

Graph of the Sine Function

To sketch the graph of y = sin x first locate the key points.These are the maximum points, the minimum points, and the intercepts.

0-1010sin x

0x2

2

32

Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period.

y

2

3

2

22

32

2

5

1

1

x

y = sin x

6. The cycle repeats itself indefinitely in both directions of the x-axis.

Properties of Sine and Cosine Functions

The graphs of y = sin x and y = cos x have similar properties:

3. The maximum value is 1 and the minimum value is –1.

4. The graph is a smooth curve.

1. The domain is the set of real numbers.

5. Each function cycles through all the values of the range over an x-interval of .2

2. The range is the set of y values such that . 11 y

Transformations – A look backLet’s go back to the quadratic equation in

graphing form: y = a(x – h)2 + k

If a < 0: reflection across the x axis|a| > 1: stretch; and |a| < 1: shrink(h, k) was the vertex (locator point)h gave us the horizontal shiftk gave us the vertical shift

Transforming the Cosine (or Sine)y = a (cos (bx – c)) + d

Let’s look at a|a| is called the amplitude, like our other

functions it is like a stretchIf a < 0 it also causes a reflection across the

x-axisGraph

y = cos x and y = 3 cos xy = cos x and y = -3 cos x

Key PointsAmplitude – increase the output by a factor of

the amplitudeRemember the amplitude is always positive so

you have to apply any reflectionsy = 3 cos x

30-303cos x

0x2

2

32

10-101cos x

0x2

2

32

y

1

123

2

x 32 4

Example: Sketch the graph of y = 3 cos x on the interval [–, 4].

Partition the reference interval [0, 2] into four equal parts. Find the five key points; graph one cycle; then repeat the cycle over the interval.

maxx-intminx-intmax30-303y = 3 cos x20x 2

2

3

(0, 3)

2

3( , 0)( , 0)

2

2( , 3)

( , –3)

The amplitude of y = a sin x (or y = a cos x) is half the distance between the maximum and minimum values of the function. amplitude = |a|

If |a| > 1, the amplitude stretches the graph vertically.If 0 < |a| < 1, the amplitude shrinks the graph vertically.If a < 0, the graph is reflected in the x-axis.

2

32

4

y

x

4

2

y = – 4 sin xreflection of y = 4 sin x y = 4 sin x

y = 2 sin x

2

1y = sin x

y = sin x

Transforming the Cosine (or Sine)y = a (cos (bx – c)) + d

Let’s look at dJust like in our other functions, d is the

vertical shift, if d is positive, it goes up, if it is negative, it goes down.

Graph y = cos x and y = cos x + 1y = cos x and y = cos x – 2

Vertical Shifty = cos x + 1Begin with y = cos x Add d to the output to adjust the graph

Then shift up one unit

y

2

3

2

22

32

2

5

1

1

x

10-101cos x

0x2

2

32

21012cos x

0x2

2

32

Transforming the Cosine (or Sine)y = a (cos (bx – c)) + d

Let’s look at c Just like in our other functions, c is the

horizontal shift, if c is positive, it goes right, if it is negative, it goes left. If there is no b present, it is the same as other functions

Graph:y = cos x and y = cos (x + π/2)y = cos x and y = cos (x - π/2)

Horizontal shifty = cos (x + π/2)

Begin with y = cos xNow you must translate the input… the

angle

Then shift left π/2 units

y

2

3

2

22

32

2

5

1

1

x

10-101cos x

0x2

2

32

10-101cos x

0x2

2

32

Transforming Cosine (or Sine)

y = a (cos (bx – c)) + d

Let’s look at b

Graph:y =cos x and y =cos 2x (b = 2)

y =cos x and y =cos ½ x (b = ½ )

What happened?

Transforming the Periodb has an effect on the period (normal is 2π)If b > 1, the period is shorter, in other words,

a complete cycle occurs in a shorter interval

If b < 1, the period is longer or a cycle completes over an interval greater than 2π

To determine the new period = 2π/b

y

x

2

sin xy period: 2 2sin xy

period:

The period of a function is the x interval needed for the function to complete one cycle.

For b 0, the period of y = a sin bx is .b

2

For b 0, the period of y = a cos bx is also .b

2

If 0 < b < 1, the graph of the function is stretched horizontally.

If b > 1, the graph of the function is shrunk horizontally.

y

x 2 3 4

cos xy period: 2

2

1cos xy

period: 4

Summarizing …Standard form of the equations: y = a (cos (bx – c)) +

d y = a (sin (bx – c)) + d

“a” - |a| is called the amplitude, like our other functions it is like a stretch it affects “y” or the outputIf a < 0 it also causes a reflection across the x- axis

“d” – vertical shift, it affects “y” or the output“c” – horizontal shift, it affects “x” or “θ” or the input“b” – period change (“squishes” or “stretches out” the

graphThe combination of “b” and “c” has another effect that

we will discuss next time.

Next classWe will get into the detail of transforming

the period when both b and c are presentWe will graph using a “key point” methodWe will talk about how the calculator can

help and how you need to be careful with setting windows.

Homework 24Section 4.5, p. 3073-21 odd, 23-26 all