13.5 – The Cosine Function

The Cosine FunctionFind the x-coordinate of each point on the unit circle.

1. A 2. B

3. C 4. D

The Cosine Function1. x-coordinate of point A: 1

2. x-coordinate of point B: 0

3. x-coordinate of point C: –1

4. x-coordinate of point D: 0

Solutions

The Cosine Function

The

Highlights of the Cosine Function• The cosine function, matches the measure of an

angle in standard position with the x-coordinate of a point on the unit circle.

• Within one cycle of the function the graph will “zero” by touching the x axis two times ( ); reach a minimum value of -1 at and a maximum value of 1 at 0 and .

cosy

2

3,2

2

Differences and Similarities between the sine and cosine functions

Graphing the Cosine Function Sketch the graph of

4cos2

y

Steps:

1. Determine the amplitude. In this case a = 2.

2. Determine the period using the formula . This will be the outer boundary of your graph.

Period =

3. Use five points equally spaced through one cycle to sketch a cosine curve. The five–point pattern is

max–zero–min–zero-max.Plot the points.

b

2

84

1

2

4

2

8

Graphing the Cosine Function Sketch the graph of

4cos2

y

Steps:

4. Make a smooth curve through the points to complete your graph.

8

The Cosine FunctionUse the graph shown below.

a. Find the domain, period range, and amplitude of this function.

The domain of the function is all real numbers.

The function goes from its maximum value of 2 and back again in an interval from 0 to 2 . The period is 2 .

The amplitude is (maximum – minimum) = (2 – (–2)) = (4) = 2.12

12

12

The function has a maximum value of 2 and a minimum value of –2. The range is –2 y 2.<– <–

The Cosine Function(continued)

The maximum value occurs at 0 and 2 .

b. Examine the cycle of the cosine function in the interval from 0 to 2 . Where in the cycle does the maximum value occur? Where does the minimum occur? Where do the zeros occur?

The minimum value occurs at .

The zeros occur and at .2 23

Divide the period into fourths. Plot five points for the first cycle. Use 2 for the maximum and –2 for the minimum.

Repeat the pattern for the second cycle.

The Cosine Function Sketch the graph of y = –2 cos in the interval from 0 to 4.

| a | = 2, so the amplitude is 2.

b = b , so the graph has 2 full cycles from 0 to 4. = 2, so the period is 2.2

Sketch the curve.

The Cosine FunctionSuppose 8-in. waves occur every 6 s. Write an equation that

models the height of a water molecule as it moves from crest to crest.

= 4 Simplify.

a = amplitude = 82

maximum – minimum2

period = Use the formula for the period.2b

6 = The period is 6. Substitute.2b

b = Multiply each side by .b62b

= Simplify.3

The equation will have the form y = a cos b . Find the values for a and b.

An equation that models the height of the water molecule is y = 4 cos . 3

Step 1: Use two equations. Graph the equations y = 1 and y = –2 cos on the same screen.2x

3

The Cosine FunctionIn the function y = –2 cos , for which values of x is the

function equal to 1?

2x3

Solve the equation, 1 = – 2 cos , for the interval of 0 to 10.2x3

The graph show two solutions in the interval. They are x = 3.14 and 6.28.

Step 2: Use the Intersect feature to find the points at which the two graphs intersect.

The solution to the equation for the interval 0 x 10 is 3.14 and 6.28, or and 2 .

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