Name: ______Date: ______Hr: _____
Trigonometric Models WS #1
Graphing Sine & Cosine Using Radians
1. Fill in the degree equivalent of the angles below, which have been given in radians. Use this chart as a reference for the most common angles in radians.
q (radians) / 0 / p6 / p
4 / p
3 / p
2 / 2p
3 / 3p
4 / 5p
6 / p / 5p
4 / 3p
2 / 7p
4 / 2p
q (degrees)
2. Now graph y = sin x where the angle x is given in radians rather than degrees. To graph by hand, it will be necessary to write the radian measure as a decimal (to accurately locate it on the x-axis). You may also fill in the value of sin x as a decimal. If you put your calculator in “RADIAN” mode, it will get the proper value for
sin x when your angle is entered in radians rather than degrees.
(a) Fill in the chart below and then use the values to complete the graph. One row has been done for you.
x (radians) / x(decimal) / y = sin x / x (radians) / x
(decimal) / y = sin x
0 / 5p
6
p
6 / 0.52 / 0.5 / p
p
4 / 5p
4
p
3 / 3p
2
p
2 / 7p
4
2p
3 / 2p
3p
4
(b) Last week you graphed y = sin x where angle x was measured in degrees rather than radians. Look back at the last few pages of Ch 6 WS #1 and find this graph. How is it the same as the graph in radians? How is it different?
3. Now graph y = cos x where the angle x is given in radians rather than degrees.
(a) Fill in the chart below and then use the values to complete the graph. One row has been done for you.
x (radians) / x(decimal) / y = cos x / x (radians) / x
(decimal) / y = cos x
0 / 5p
6
p
6 / 0.52 / 0.87 / p
p
4 / 5p
4
p
3 / 3p
2
p
2 / 7p
4
2p
3 / 2p
3p
4
(b) Last week you graphed y = cos x where angle x was measured in degrees rather than radians. Look back at the last few pages of Ch 6 WS #1 and find this graph. How is it the same as the graph in radians? How is it different?
3. Functions that repeat themselves over and over again (like the sine and the cosine graphs) are called PERIODIC FUNCTIONS. The PERIOD of such functions is the amount of time (along the x-axis) it takes for the function to complete one full cycle. Using your graphs above and those from WS #1:
(a) What is the PERIOD of the graph of y = sin x when you graph in radians? What is the PERIOD of the graph of y = sin x when you graph in degrees?
(b) What is the PERIOD of the graph of y = cos x when you graph in radians? What is the PERIOD of the graph of y = cos x when you graph in degrees?
4. Is it possible to make a formula that CHANGES the period of sine and cosine?
(a) To experiment, fill in the chart below for y = 2∙sin x, where x is given in radians. When you have filled in the chart carefully sketch the result.
x (radians) / x(decimal) / y = 2∙sin x / x (radians) / x
(decimal) / y = 2∙sin x
0 / 5p
6
p
6 / 0.52 / 1.0 / p
p
4 / 5p
4
p
3 / 3p
2
p
2 / 7p
4
2p
3 / 2p
3p
4
(b) Is the PERIOD of this new graph of y = 2∙sin x the same or different than the PERIOD of the graph of y = sin x as graphed in question #2? If the period is different, describe how. If the period is NOT different, explain what IS different about this graph.
5. Is there another way to make a formula that CHANGES the period of a sine or cosine graph?
(a) To experiment, fill in the chart below for y = sin (2x), where x is given in radians. When you have filled in the chart carefully sketch the result.
x (radians) / x(decimal) / y = sin(2x) / x (radians) / x
(decimal) / y = sin(2x)
0 / 5p
6
p
6 / 0.52 / 0.87 / p
p
4 / 5p
4
p
3 / 3p
2
p
2 / 7p
4
2p
3 / 2p
3p
4
(c) Is the PERIOD of this new graph of y = sin(2x) the same or different than the PERIOD of the graph of y = sin x as graphed in question #2? If the period is different, describe how. If the period is NOT different, explain what IS different about this graph.
6. Recall back to chapter 3 when you had to make new graphs from some well-known Parent Functions. Suppose you have a parent function f(x) = x2 and you wanted to alter it by changing the constants A, B, C, and D such that the new function could be in the form A∙f(Bx + C) + D.
(a) What happened to the graph of f(x) = x2 if a constant was added to the output, for example, if you graphed f(x) = x2 + 3? Would this be changing A, B, C, or D?
(b) What happened to the graph of f(x) = x2 if a constant was added to the input, for example, if you graphed
f(x) = (x + 3)2 ? Would this be changing A, B, C, or D?
(c) What happened to the graph of f(x) = x2 if a constant was multiplied by the input, for example, if you graphed
f(x) = (2∙x)2 ? Would this be changing A, B, C, or D?
(d) What happened to the graph of f(x) = x2 if a constant was multiplied by the output, for example, if you graphed
f(x) = 2∙x2 ? Would this be changing A, B, C, or D?
(e) If your parent function is NOW f(x) = sin x, which constant are you changing when you graph f(x) = 2∙sin x? Which constant are you changing when you graph f(x) = sin(2∙x)? How did these changes effect the graph?