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The Graphs of the Sine and Cosine Functions

By Jonathan Freeman,2014-06-17 21:55
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The Graphs of the Sine and Cosine Functions

Graphs of the Sine and Cosine Functions

Review of Prerequisites:

1. The unit circle is the circle that has as its center the origin of the coordinate plane and a radius

of length one. Using a rectangular coordinate system, the unit circle is the set of points 22described by {(x,y) | x + y = 1}. Using a polar coordinate system, the unit circle is described

by {(r,;) | r = 1}.

2. The wrapping function is the function P that assigns to each number t on the number line,

the ordered pair representing the coordinates of the point on the unit circle to which it is

attached when the number line is wrapped around the unit circle as described: 0 is attached to

the point (1,0); the positive half of the number line is then wrapped around the unit circle in

the counterclockwise direction and the negative half wrapped in the clockwise direction.

The abscissa of the point on the unit circle to which the point t on the number line has been

attached is defined to be the cosine of t, written cos t. The ordinate of this point is defined to

be the sine of t, written sin t.

3. Equivalently, for any real number t, the cosine of t (cos t) and the sine of t (sin t) may be

defined as the first and second coordinates respectively of the image of the point (1,0) under a

rotation of magnitude t about the origin.

Objectives:

1. To use the wrapping function, the Pythagorean Theorem, and the symmetries of the unit circle

to calculate exact values of the sine and cosine functions of a selected set of numbers.

2. To graph the sine and cosine functions.

Materials:

1. TI82 Calculator

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Activity:

1.. Use the following diagrams, the Pythagorean Theorem, and the geometry of the circle to find

the exact values of the sine and cosine functions indicated in the tables that follow the

diagrams. Use your calculator to approximate t, sin t, and cos t each to two decimal places. 1

a.

(0,1)

;P4

1

(-1,0)(1,0)

(0,-1)

In the diagram above, label the lengths of the legs of the right triangle that is shown. Discuss

the geometric concepts that are used in finding these lengths.

List two additional values of t that the wrapping function P assigns to the point P(/4).

2

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b.

(0,1)

;P6

1

(-1,0)(1,0)

(0,-1)

In the diagram above, label the lengths of the legs of the right triangle that is shown. Discuss

the geometric concepts that are used in finding these lengths.

List two additional values of t that the wrapping function P assigns to the point P(/6).

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c.

0 /6 /4 /3 /2 2/3 3/4 5/6 Exact t

Approx.

Exact cos t

Approx.

Exact sin t

Approx.

What symmetry of the unit circle may be used to obtain the values of sin t and cos t for

values of t in the table that are larger than /2 from those corresponding to values of t

smaller than /2 ? Explain.

d.

0 -/6 -/4 -/3 -/2 -2/3 -3/4 -5/6 - Exact t

Approx.

Exact cos t

Approx.

Exact sin t

Approx.

What symmetry of the unit circle may be applied to obtain the values in this table from those

in (c)? Explain.

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e.

7/6 5/4 4/3 3/2 5/3 7/4 11/2 Exact t 6

Approx.

Exact cos t

Approx.

Exact sin t

Approx.

What symmetry of the unit circle may be applied to obtain the values in this table from those in

(c)? Explain.

4. On the coordinate axes provided, approximate the graphs of y = cos t between t = -2 and

t = 2 by plotting the points in the preceding tables, and drawing a smooth curve through the

points.

y = cos t

1

0.5

-0.5

-1

3

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5. On the coordinate axes provided, approximate the graphs of y = sin t between t = -2 and

t = 2 by plotting the points in the preceding tables, and drawing a smooth curve through the

points.

y = sin t

1

0.5

-0.5

-1

4

6. What do you observe about the graph of each from t = -2 and t = 0 as compared with the

graph from t = 0 to t = 2? Explain why this is the case.

7. How would you describe the graph of each function from t = 2 to t = 4? How would

you extend the graphs to include all values of t on the number line? Explain how, and why

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8. What, if any, symmetries do each of these graphs possess? Explain.

9. If you know that the sine of a number is 0.7, can you find its cosine? Explain how and why.

10. A toddler's swing set is constructed with the seat supported by 6' metal rods that are restricted

from rotating more than 45 from the vertical position. The seat is 2' above the ground. How

high above the ground is it possible for an energetic child to swing? (Use the ideas in this

activity, circles and the concept of similarity to solve this problem.)

Sine and Cosine Graphs

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