Download Graphing of Trig Functions 2

Graphing of Trigonometric Functions Lesson Plan
By: Douglas A. Ruby
Class: Pre-Calculus II
Date: 10/10/2002
Grades: 11/12
INSTRUCTIONAL OBJECTIVES:
At the end of this lesson, the student will be able to:
1. Evaluate trigonometric functions with the TI-83+ using either degree or radian measure.
2. Use a TI-83+ Graphing Calculator to graph all six basic trigonometric functions in the form
of y = sin(x) , y = cos(x), y=yan(x), y=sec(x), y=csc(x), and y=cot(x).
3. Given a trigonometric function in the form of y=sin(x), y=cos(x), etc., be able to sketch the
curve.
4. Use the ZOOM, WINDOW, and TRACE functions to evaluate the trigonometric functions
for various angles.
Relevant Massachusetts Curriculum Framework
PC.P.3 - Demonstrate an understanding of the trigonometric functions (sine, cosine, tangent,
cosecant, secant, and cotangent). Relate the functions to their geometric definitions.
MENTAL MATH – (5 Minutes)
Question 1: Recall the discussion on trigonometric functions of a right triangle discussed
yesterday. What does the acronym SOHCAHTOA stand for?
Question 2: What are the values for the six trigonometric functions for the following angles:
= /3
Sin =
Cos =
Tan =
= 11 /4
Csc =
Sec =
Cot =
Sin =
Cos =
Tan =
Page 1
Csc =
Sec =
Cot =
Graphing Trigonometric Functions using a TI-83+ – Mr. Ruby
CLASS ACTIVITIES – (Note: 45 Minute Lesson Plan)
1. Basics of using the TI-83+ to graph trigonometric functions.
Ok, we are going to start this class by using the TI-83+ to graph the basic trigonometric functions
of sin, cos, tan, csc, sec, and cot. To start with, let’s look at the basic TI-83+ keyboard. Your
keyboard should look like this:
Arrow Keys
Start by hitting the ON key at the lower left of the keypad. Next, let’s set ourselves up to graph
trigonometric functions using degree measurement. To do this, let’s hit the MODE key next to
the 2nd key. You should see the following window:
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Graphing Trigonometric Functions using a TI-83+ – Mr. Ruby
Use the Down Arrow (the purple one in the upper right quadrant) to position the blinking cursor
to “radians”. Next, use the Right Arrow to move the cursor to “degrees” and then hit Enter
followed by Clear. We now have set the TI-83+ up so that the “x-axis” and any angular data we
enter are assumed to be in units of degrees. We now want to set up our display window so that
the vertical y-axis is 5 units (+ or -) and the horizontal axis is 360 degrees plus and minus.
To do this, hit the WINDOW key next to the Y= key. You should see:
Let’s change the display windows size so that Xmin=-360, Xmax=+360, Ymin=-5, and Ymax=5.
Do this by using the Arrow Keys to position the cursor and overtyping the previous numbers.
Use DEL to clear unwanted characters. When you are done, this window should look like this:
Ok, now we are ready to start graphing. Hit the Y= key to get this window:
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Graphing Trigonometric Functions using a TI-83+ – Mr. Ruby
Now we will enter our first trigonometric function. Please enter sin(x) by hitting the SIN key,
followed by the X,T, ,n key, followed by a closed parentheses “)”. Your screen should look like:
Now, let’s graph the y = sin x function by hitting the GRAPH key. Your screen should look like:
Use the Arrow Keys to move the arrow around the screen. See if you can find the value of x
when y=1? What do you get? Try following the curve and see if you can find some values for
y=sin x?
Notice that it is difficult to get exactly correct values? Hit the CLEAR button twice to get back
to the basic entry screen. Let’s calculate the value of y=sin(67.5o). You should see:
Now go back and try to find y=sin(67.5o) by tracing the graph from before. Hit the GRAPH
button again. Now hit the TRACE button. Notice that the trace cursor is at x=0 and y=0. Move
the trace cursor to the right until x=67.5. When I try to do this, I can only get as accurate as
x=68.93617 and y=.96774194. However, we can ZOOM in to see this part of the curve more
closely. Let’s hit the ZOOM button and ZOOM IN (followed by ENTER). You now should
see:
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Graphing Trigonometric Functions using a TI-83+ – Mr. Ruby
Let’s continue to TRACE and see if we can find y=sin(67.5o). I still have a hard time getting
exactly x=67.5o. Let’s now try a different form of ZOOM. Hit ZOOM and then hit ZTRIG
followed by ENTER. You will see that graph looks pretty good now. When you TRACE, you
will see that the cursor moves in units of 7.5o. This is nice, because we can look at the y value of
y=sin(x) for x=7.5o, 15o, 22.5o, 30o, 37.5o, 45o, 52.5o, 60o, etc.
By ZOOMING IN or ZOOMING OUT we can also move around in the curve and find other
(x,y) values for y=sin(x). Of course, we don’t need our calculators for the 30o, 45o,60o, 90o
angles, since we already have those committed to memory, right!
Right now, I’d like you to try graphing y=cos(x), use the ZTRIG, ZOOM IN, and ZOOM OUT
to focus on different parts of the curve. Using the trace function, please find the following:
a) y = cos (67.5o)
.382
e) y = tan (67.5o)
2.41
b) y =cos (367.5o)
.991
f) y = tan (367.5o)
.131
c) y =cos (-52.5o)
.609
g) y = tan (-52.5o)
-1.30
d) y = cos (-65.625o)
.413
f) y = tan (-65.625o)
-2.207
2. Graphing all six trigonometric functions
Ok, now we will graph all six trig functions. Notice from the keypad, that we only have SIN,
COS, and TAN keys. To graph the secant, cosecant, and cotangent functions, what will have to
do? Now try graphing y=sec(x). First, set up your ZOOM to ZTRIG and hit ENTER. Now, hit
the Y= key and clear any previous entries by positioning the cursor to the entry using the
ARROW KEYS and then hitting CLEAR. Now enter y=sec(x). How do we do this? Well, do
you remember that sec(x) = 1/cos(x)? Given that we know the definitions of all six trigonometric
functions, we can use the COS(X) key to create the SEC(X) function by setting Y1=1 COS(x).
You should see the following:
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Graphing Trigonometric Functions using a TI-83+ – Mr. Ruby
Now, we can graph the y=sec(x) function and we should see:
Try tracing this function to values of x=90o. Notice that at X=90o, Y=<blank>. Why is this so?
This is because cos(90o) = 0, so 1 cos(90o) = sec(90o) is undefined.
Now go ahead and graph the cosecant and cotangent functions. To do this, please again use the
zooming and trace functions to find:
a) y = csc (67.5o)
1.082
e) y = cot (67.5o)
.414
b) y = sec (367.5o)
1.009
f) y = csc (367.5o)
7.661
c) y = 1/tan (-52.5o)
-.767
g) y = cos (-52.5o)
.609
d) y = csc (-65.625o)
-1.098
f) y = sec (-65.625o)
2.423
You now know how to graph the basic six trigonometric functions. Before we finish today’s
lesson, let’s also look at two additional topics.
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Graphing Trigonometric Functions using a TI-83+ – Mr. Ruby
3. Graphing using Radian instead of Degree measure
In order to use radians instead of degrees, let’s change our calculator MODE. First, go back to
Y= and make sure Y1=sin(x). Then hit the MODE button and you should see:
Use the arrow keys to highlight Radian instead of Degree then hit ENTER. If you now return to
the GRAPH function, you should see:
Let’s reset the ZOOM to ZTRIG. Now TRACE the graph. You should notice that at
x=1.5707963, y=1. This value x=1.5707963) is, in fact, an approximation for x= /2 radians. If
you have done this correctly, your screen should look like:
To approximate when tracing radians, use =3.14. Therefore, a decimal approximations of /6
is 3.14 6 = .523. To approximate and (n x /6), you would multiply .512 by n. The same can be
done for multiples of /2 = 1.57, /4 = .785, and /3 = 1.046. Now, use the trace functions and
radian measure to find the following:
a) y = csc ( /2)
1
e) y = cot (3.2)
17.102
b) y = sec (3 /2)
c) y = cot (- /6)
(undefined) -2.95*10-8
-1.732
f) y = tan ( /3)
g) y = cos (13 /2)
1.732
(0) 1.3*10-9
Page 7
d) y = cos (11 /6)
.866
f) y = sec (-2 )
1
Graphing Trigonometric Functions using a TI-83+ – Mr. Ruby
Note: Some value should be undefined or zero but may not be due to numerical error.
Summary: We have covered using a TI-83+ calculator to evaluate and graph trigonometric
functions. Tonight’s homework asks you to continue this process. You will evaluate functions
using the normal mode as well as by tracing the graph of the function. You will be asked to
sketch graphs of all six functions. You will also be asked to look at the graph of a “constant”
times a trigonometric function in the form of y = ½ * cos(x) and compare it with y = cos(x).
When you compare these two graphs, I’d like you to see what you can learn about the impact of
the constant on the shape of the graph. Does the constant “½ “change the period (cyclical nature)
or amplitude (height of rise and fall) of the curve?
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Graphing Trigonometric Functions using a TI-83+ – Mr. Ruby
HOMEWORK:
1. Use the regular calculator functions to evaluate y=tan(x) at:
a. 120o
b. /2 radians
c. 13o
2. Graph the y=tan(x) functions, use trace and zoom to evaluate the function at:
a. 120o
b. /2 radians
c. 13o
3. Using the graphing calculator to guide you, sketch graphs of the six trigonometric functions
over the domain x=-2 to x=2 radians (-360o to 360o). Label each graph with the name of
the function. Label the x-axis with degrees or radians and label the y-axis with the correct y
values.
Sine
Cosecant
Cosine
Secant
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Tangent
Cotangent
Graphing Trigonometric Functions using a TI-83+ – Mr. Ruby
4. Sketch a graph of y=cos(x) for x= -2 to 2 radians. Now, sketch a graph of y= ½*cos(x) on
top of the first graph. Use the graph box below.
5. Now use your calculator to graph the two functions in #4 above. Graph both functions on the
same screen. What is the period of y=cos(x)? What is the amplitude of y=cos(x? What is the
period of y= ½*cos(x)? What is the amplitude of y= ½*cos(x)?
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Graphing Trigonometric Functions using a TI-83+ – Mr. Ruby
Answer Keys:
MENTAL MATH – (5 Minutes)
Question 1: Recall the discussion on trigonometric functions of a right triangle discussed
yesterday. What does the acronym SOHCAHTOA stand for?
For any angle of a right triangle, the:
Sine of = Opposite side over Hypotenuse side
Cosine of = Adjacent side over Hypotenuse side
Tangent of = Opposite side over Adjacent side
Question 2: What are the values for the six trigonometric functions for the following angles:
= /3 (60o)
Csc
=2
=12
Sec
=2
=
Cot
=1
Sin
=
Cos
Tan
3 2
= 11 /4 (2 + 3 /4 or 360o + 135o)
3
3
3
Sin
=1
Cos
= 1
Tan
= -1
Page 11
2
2
Csc
=
Sec
=
Cot
= -1
2
2
Graphing Trigonometric Functions using a TI-83+ – Mr. Ruby
HOMEWORK (Answer Key):
1. Use the regular calculator functions to evaluate y=tan(x) at:
a. 120o = -1.732050808 (or - 3 )
b. /2 radians = get ERR:DOMAIN since tan( /2) is undefined
c. 13o = .2308681911
2. Graph the y=tan(x) functions, use trace and zoom to evaluate the function at:
d. 120o = -1.732051
e.
/2 radians = -4.8757x10-9
f. 13o = .23086819
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Graphing Trigonometric Functions using a TI-83+ – Mr. Ruby
3. Using the graphing calculator to guide you, sketch graphs of the six trigonometric functions
over the domain x=-2 to x=2 radians (-360o to 360o). Label each graph with the name of
the function. Label the x-axis with degrees or radians and label the y-axis with the correct y
values. (Note: x range is [-2 , 2 ] in all six graphs, y range is specified in title.)
Sine (y=[-2,2])
Cosecant (y=[-6,6])
Cosine (y=[-2,2])
Secant (y=[-6,6])
Tangent (y=[-4,4])
Cotangent (y=[-4,4])
4. Sketch a graph of y=cos(x) for x= -2 to 2 radians. Now, sketch a graph of y= ½*cos(x) on
top of the first graph. Use the graph box below.
5. Now use your calculator to graph the two functions in #4 above. Graph both functions on the
same screen. What is the period of y=cos(x)? What is the amplitude of y=cos(x? What is the
period of y= ½*cos(x)? What is the amplitude of y= ½*cos(x)?
See screen shot above. Note that period of y=cos(x) is still 360o. Amplitude of wave form is
now ½ instead of 1.
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Graphing Trigonometric Functions using a TI-83+ – Mr. Ruby
Powerpoint Slides to go with Lesson:
Objectives
At the end of this lesson, the student
will be able to:
Evaluate trigonometric functions with the TI83+ using either degree or radian measure.
Use a TI-83+ Graphing Calculator to graph all
six basic trigonometric functions in the form of
y = sin(x) , y = cos(x), y=yan(x), y=sec(x),
y=csc(x), and y=cot(x).
Given a trigonometric function in the form of
y=sin(x), y=cos(x), etc., be able to sketch the
curve.
Use the ZOOM , WINDO W , and TRACE
functions to evaluate the trigonometric
functions for various angles.
Mental Math
Question 1: Recall the discussion on
trigonometric functions of a right triangle
discussed yesterday. What does the
acronym SOHCAHTOA stand for?
S________ = O_______/H_____________
C________= A________/H____________
T________= O________/A____________
Question 2: What are the values for the six
trigonometric functions for the two angles:
= /3 (60 o)
Sin =
Cos =
Tan =
Csc
Sec
Cot
= 11 /4 (495 o)
=
=
=
Sin =
Cos =
Tan =
Page 14
Csc
Sec
Cot
=
=
=
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