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PROJECT 2

Graphs of the Trigonometric Functions

You and your group will need to complete the problems on the following pages using your unit circle, special triangles, and graphing calculator.

The graph of the Sine function

1. Consider the ordered pair:

(angle in radian measure, the sine of that angle) or [pic] where ( is measured in radians.

Using your unit circle and special triangles, fill in the values of the following chart with these ordered pairs:

|( |sin ( | |( |sin ( |

|[pic] | | | | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic][pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|0 | | |[pic] | |

2. Plot the ordered pairs from the chart on page 1 on the graph paper and axes below. Connect the points with a smooth curve. Make sure your units and axes are clearly labeled and each unit has equal measure.

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3. What is the highest output of the curve?

4. What is the lowest output of the curve?

5. What is the amplitude of the sine function? (Note: The amplitude of the sine function is defined as the distance from the midline to the highest or lowest point.)

6. The sine function is usually given by the notation [pic], where ( is measured in radians only. Put your calculator in radian mode, graph the sine function on your calculator, and compare it to your graph above. (Note: use the window [pic] with [pic]). Does the calculator graph appear to be the same as your graph?

7. Does the sine function have repeating outputs? If so, what is the period of the outputs, hence the period of the sine function? (Note: The period of a function is the length of the interval on the x-axis where the graph repeats the same outputs over and over again.)

8. Does the sine function appear to be symmetric about the y-axis or the origin? If so, which?

9. Using the graph on your calculator, find all the solutions to the equation [pic]. (Hint: Remember finding the solutions to [pic] is the same as finding all the zeros, roots, or x-intercepts of f(x).)

10. List the transformations needed to change the graph of [pic] into the graph of [pic].

The graph of the Cosine function

We create the graph of the cosine function the same way we do for the sine function, using the unit circle and special triangles. To avoid unnecessary work, answer the following questions about the cosine function from the graph on your calculator.

1. The cosine function is usually given by the notation [pic], where ( is measured in radians only. Put your calculator in radian mode, graph the cosine function on your calculator, and copy this graph onto your graph paper below labeling everything very clearly. (Note: use the window [pic] with [pic]).

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| | | | | | | | | | | | | | | | | | | | | | | | |

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2. What is the highest output of the curve?

3. What is the lowest output of the curve?

4. What is the amplitude of the cosine function? (Note: The amplitude of the cosine function is defined as the distance from the midline to the highest or lowest point.)

5. Does the cosine function have repeating outputs? If so, what is the period of the outputs, hence the period of the cosine function? (Note: The period of a function is the length of the interval on the x-axis where the graph repeats the same outputs over and over again.)

6. Does the cosine function appear to be symmetric about the y-axis or the origin? If so, which?

7. Find all the solutions to the equation [pic].

8. Find all solutions in the interval [pic] to the equation [pic].

9. List the transformations needed to change the graph of [pic] into the graph of [pic]

10. List the transformations needed to change the graph of [pic] into the graph of [pic]

11. Using the graphs of [pic] and [pic], answer the following questions.

a. By transforming the graphs of sine and cosine, is [pic]true, hence yielding a trigonometric identity?

b. By transforming the graphs of sine and cosine, is [pic]true, hence yielding a trigonometric identity?

The graph of the Tangent function

1. We create the graph of the tangent function the same way we do for the sine and cosine functions, using the unit circle and special triangles. Using your unit circle and special triangles, fill in the values of the following chart.

|( |tan ( | |( |tan ( |

|[pic] | | | | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic][pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|0 | | |[pic] | |

2. Plot the ordered pairs from the chart on the previous page on the graph paper and axes below. Connect the points with a smooth curve. Make sure your units and axes are clearly labeled and each unit has equal measure.

| | | | | | | | | | | | | | | | | | | | | | | | |

| | | | | | | | | | | | | | | | | | | | | | | | |

| | | | | | | | | | | | | | | | | | | | | | | | |

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3. What is the highest output of the curve?

4. What is the lowest output of the curve?

5. Does the tangent function have finite amplitude? If it does not, then we say it has no amplitude.

6. Does the tangent function have repeating outputs? If so, what is the period of the outputs, hence the period of the tangent function? (Note: The period of a function is the length of the interval on the x-axis where the graph repeats the same outputs over and over again.)

7. Does the tangent function appear to be symmetric about the y-axis or the origin? If so, which?

8. Find all the solutions on the interval [pic] to the equation [pic].

9. Find [pic] where the terminal side of an angle of ( radians lies on the line [pic].

10. There are parts of the tangent function that are undefined.

a. For what values of x is the tangent function undefined on the interval [pic]?

b. How far apart are the points where the tangent function is undefined? (Note: This number should match the number you got as the period.)

The graph of the Secant function

The secant function is defined by assigning to the angle ( the reciprocal of the x-coordinate of the ordered pair where the terminal side of the angle ( crosses the unit circle (e.g. [pic]).

1. Using the definition of secant, fill in the following chart with the output for the secant function.

|( |sec ( | |( |sec ( |

|[pic] | | | | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic][pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|[pic] | | |[pic] | |

|0 | | |[pic] | |

2. Plot the ordered pairs from the chart on the previous page on the graph paper and axes below. Connect the points with a smooth curve. Make sure your units and axes are clearly labeled and each unit has equal measure.

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| | | | | | | | | | | | | | | | | | | | | | | | |

| | | | | | | | | | | | | | | | | | | | | | | | |

3. Does the secant function have an amplitude? Why or why not?

4. Is the secant function periodic? If so, what is its period?

5. Are there any asymptotes? If so, where and how often?

6. Would you say that secant function has any symmetry? If so, what kind?

7. Using a dotted line, sketch the graph of the cosine function on the same set of axes you sketched the secant function.

a. What do you notice about the graphs?

b. How are these functions related?

The graph of the Cosecant function

The cosecant function is defined by assigning to the angle ( the reciprocal of the y-coordinate of the ordered pair where the terminal side of the angle ( crosses the unit circle (e.g. [pic]).

1. Using what you know about the relationship between the secant and cosine functions and their graphs, sketch the graph of the cosecant function using the sine function below. Have ( go from -2( to 2( labeling all standard angles and their outputs.

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2. If you wanted to check your graph on your calculator, what equation would you enter into Y1?

3. Does the cosecant function have an amplitude? Why or why not?

4. Is the cosecant function periodic? If so, what is its period?

5. Does the cosecant function have any symmetry? If so, what kind?

6. Does the cosecant function have any asymptotes? If so, where are they and how often do they occur (especially in regard to the sine function)?

The graph of the Cotangent function

The cotangent function is defined as the reciprocal function of the tangent function. (Note: The notation is [pic].)

1. Using your calculator having the x-scale set on the interval [-2(, 2(] achieve a graph for the cotangent function and sketch it very precisely below labeling all standard angles and their outputs clearly.

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2. What equation did you enter into Y1 and why did you choose that?

3. Describe how the graph of the cotangent function differs from that of the tangent function being careful to include a discussion on amplitude, period, asymptotes, general behavior (i.e. increasing, decreasing, etc.), symmetry and anything else that would yield a clear picture of the properties of the cotangent function.

On a separate paper complete the following problems using your knowledge about the unit circle and special triangle, the definitions of the six trigonometric functions and their identities.

1. [pic] and [pic]

2. [pic] and [pic]

3. [pic] and [pic]

4. [pic] and [pic]

5. [pic] and [pic]

6. [pic] and [pic]

7. [pic] and [pic]

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[pic]

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