Name: ____________________________
Ch 6 WS #5
Date: ______________
Hr: _____
Graphing Other Trigonometric Functions
Press the [MODE] key on your calculator and make sure you are in RADIANS before you
complete the following.
Graphing y = tan(x):
Press the [Y=] key on your graphing calculator and enter the formula
Y1 = tan(x). For this exercise, it is better if the graph does not
connect the dots. To do this, arrow ALL THE WAY TO THE LEFT of
the Y1 formula and keep pressing [ENTER] until it shows a dotted
line. (See the picture at right.) Then choose [ZOOM] and arrow down
to 7:ZTrig and press [ENTER] to select the trigonometry window.
1. Sketch the graph of y = tan(x) below. Please label the axes. A
few of the labels have been started for you.
(a)
Where are the x-intercepts of y = tan(x)? List
the values in terms of .
(b)
What is the y-intercept of y = tan (x)?
(c)
What is happening at x = -3/2, /2, /2, and
3/2? Why does this happen?
(d)
Do you think y = tan (x) is periodic? If yes, what is its period?
(e)
Recall that the domain is the set of all possible input values for a function (usually along
x-axis) and the range is the set of all possible output values for a function. What are the
domain and range of y = tan(x)?
D: _______________________
R: _______________________
Graphing y = csc(x):
2. There is no key on your graphing calculator to graph the cosecant function. However,
recall that cosecant is just the reciprocal of another trigonometric function. Therefore, to
graph y = csc(x), we can graph:
Y1 = ________________
Press the [Y=] key on your graphing calculator and enter this formula. CONTINUE TO
KEEP THE GRAPH AS A DOTTED LINE!! Keep the trigonometry WINDOW as in the
previous question.
Sketch the graph of y = csc(x) below. Please label the axes. A few of the labels have been
started for you.
(a) Where are the x-intercepts of y = csc(x)? List
the values in terms of .
(b) What is the y-intercept of y = csc (x)?
(c) Does the graph of y = csc(x) have any places
where it is undefined and it shows vertical
asymptotes? Where are they?
(d) Do you think y = csc (x) is periodic? If yes, what is its period?
(e) What are the domain and range of y = csc(x)?
D: _______________________
R: _______________________
Graphing y = sec(x):
3. There is no key on your graphing calculator to graph the secant function. However, recall
that secant is just the reciprocal of another trigonometric function. Therefore, to graph y =
sec(x), we can graph:
Y1 = ________________
Press the [Y=] key on your graphing calculator and enter this formula. CONTINUE TO
KEEP THE GRAPH AS A DOTTED LINE!! Keep the trigonometry WINDOW as in the
previous question.
Sketch the graph of y = sec(x) below. Please label the axes. A few of the labels have been
started for you.
(b) Where are the x-intercepts of y = sec(x)? List
the values in terms of .
(b) What is the y-intercept of y = sec (x)?
(c) Does the graph of y = sec(x) have any places
where it is undefined and it shows vertical
asymptotes? Where are they?
(d) Do you think y = sec(x) is periodic? If yes, what is its period?
(e) What are the domain and range of y = sec(x)?
D: _______________________
R: _______________________
Graphing y = cot(x):
4. There is no key on your graphing calculator to graph the cotangent function. However,
recall that cotangent is just the reciprocal of another trigonometric function. Therefore, to
graph y = cot(x), we can graph:
Y1 = ________________
Press the [Y=] key on your graphing calculator and enter this formula. CONTINUE TO
KEEP THE GRAPH AS A DOTTED LINE!! Keep the trigonometry WINDOW as in the
previous question.
Sketch the graph of y = cot(x) below. Please label the axes. A few of the labels have been
started for you.
(d) Where are the x-intercepts of y = cot(x)? List
the values in terms of .
(b) What is the y-intercept of y = cot(x)?
(c) Does the graph of y = cot(x) have any places
where it is undefined and it shows vertical
asymptotes? Where are they?
(d) Do you think y = cot(x) is periodic? If yes, what is its period?
(e) What are the domain and range of y = cot(x)?
D: _______________________
R: _______________________
Adding Periodic Functions:
As you complete the following, keep your calculator in the Trig WINDOW. However, it is now
okay to have a connected line rather than a dotted line for the graphs.
5. (a) Sketch a graph of y = sin(x) + cos(x).
(b) Is this graph periodic? If yes, what is its period?
(c) What is the approximate amplitude of the
graph? (Use the TRACE key on your calculator
to trace to a high point and find this).
(d) How do the amplitude and period of this graph
compare to normal sin(x) or normal cos(x)?
Explain.
6. (a) Sketch a graph of y = 2sin(x) + cos(x).
(b) Is this graph periodic? If yes, what is its period?
(c) What is the approximate amplitude of the
graph? (Use the TRACE key on your calculator
to trace to a high point and find this).
(d) How do the amplitude and period of this graph
compare to normal sin(x) or normal cos(x)?
Explain.
7. (a) Sketch a graph of y = 2sin(x) + 2cos(x).
(b) Is this graph periodic? If yes, what is its period?
(c) What is the approximate amplitude of the
graph? (Use the TRACE key on your calculator
to trace to a high point and find this).
(d) How do the amplitude and period of this graph
compare to normal sin(x) or normal cos(x)?
Explain.
Multiplying Periodic Functions:
As you complete the following, keep your calculator in the Trig WINDOW. However, it is now
okay to have a connected line rather than a dotted line for the graphs.
8. (a) Sketch a graph of y = sin(x)∙cos(x).
(b) Is this graph periodic? If yes, what is its period?
(c) What is the approximate amplitude of the
graph? (Use the TRACE key on your calculator
to trace to a high point and find this).
(d) How do the amplitude and period of this graph
compare to normal sin(x) or normal cos(x)?
Explain.
9. (a) Sketch a graph of y = sin(x)∙sin(x) = (sin(x))2
(b) Is this graph periodic? If yes, what is its period?
(c) What is the approximate amplitude of the
graph? (Use the TRACE key on your calculator
to trace to a high point and find this).
(d) How do the amplitude and period of this graph
compare to normal sin(x) or normal cos(x)?
Explain.
10. (a) Sketch a graph of y = cos (x)∙cos(x) = (cos(x))2
(b) Is this graph periodic? If yes, what is its period?
(c) What is the approximate amplitude of the
graph? (Use the TRACE key on your calculator
to trace to a high point and find this).
(d) How do the amplitude and period of this graph
compare to normal sin(x) or normal cos(x)?
Explain.
11.(a) Now add the functions from questions 9 and 10
together and sketch a graph of
y = (sin(x))2 + (cos(x))2
(b) Why does this graph appear as it does?
Explain by looking and the graphs from
questions 9 and 10.
(c) Is this graph periodic? If yes, what is its period?
Summary:
12. When you add two periodic functions together, is the result still periodic (hint: look at
questions #5-7)? If yes, does the period always stay the same as the original two
functions?
13. When you multiply two periodic functions together, is the result still periodic (look at
questions #8-10)? If yes, does the period always stay the same as the original two
functions?
14. What will happen if you add a periodic function to a
function that is NOT periodic? Try it: Graph y = x +
sin(x). Is the result periodic?
15. What will happen if you multiply a periodic function with a
function that is NOT periodic? Try it: Graph y = x∙sin(x).
Is the result periodic? [Hint: Before you answer, press
ZOOM and zoom OUT a couple of times.]