Sharp EL-9600 User manual

APPLYING
USING THE
EL-9600
DAVID P. LAWRENCE
TRIGONOMETRY
SHARP
Graphing Calculator
SHARP TRIG COVER 02.2.19 10:55 AM Page 3

TRIGONOMETRY USING THE SHARP EL-9600 i
Applying
TRIGONOMETRY
using the
SHARP EL-9600
GRAPHING CALCULATOR
David P. Lawrence
Southwestern Oklahoma State University
This Teaching Resource has been developed specifically for use with the
Sharp EL-9600 graphing calculator. The goal for preparing this book was
to provide mathematics educators with quality teaching materials that
utilize the unique features of the Sharp graphing calculator.
This book, along with the Sharp graphing calculator, offers you and your
students 10 classroom-tested, topic-specific lessons that build skills.
Each lesson includes Introducing the Topic, Calculator Operations, Method
of Teaching, explanations for Using Blackline Masters, For Discussion,
and Additional Problems to solve. Conveniently located in the back of
the book are 36 reproducible Blackline Masters. You’ll find them ideal
for creating handouts, overhead transparencies, or to use as student
activity worksheets for extra practice. Solutions to the Activities are
also included.
We hope you enjoy using this resource book and the Sharp El-9600
graphing calculator in your classroom.
Other books are also available:
Applying STATISTICS using the SHARP EL-9600 Graphing Calculator
Applying PRE-ALGEBRA and ALGEBRA using the SHARP EL-9600 Graphing Calculator
Applying PRE-CALCULUS and CALCULUS using the SHARP EL-9600 Graphing Calculator
Graphing Calculators: Quick & Easy! The SHARP EL-9600

ii TRIGONOMETRY USING THE SHARP EL-9600
Dedicated to my grandma, Marjorie Lee
Special thanks to Ms. Marina Ramirez and Ms. Melanie Drozdowski for
their comments and suggestions.
Developed and prepared by Pencil Point Studio.
Copyright © 1998 by Sharp Electronics Corporation.
All rights reserved. This publication may not be reproduced,
stored in a retrieval system, or transmitted in any form or by
any means, electronic, mechanical, photocopying, recording,
or otherwise without written permission.
The blackline masters in this publication are designed to be used with
appropriate duplicating equipment to reproduce for classroom use.
First printed in the United States of America in 1998.

TRIGONOMETRY USING THE SHARP EL-9600 iii
CONTENTS
CHAPTER TOPIC PAGE
1 Graphs of Trigonometric Functions 1
2 Evaluating Trigonometric Functions 5
3 Graphical Verification of
Formulas and Identities 10
4 Amplitude, Period and Phase Shift 14
5 Inverse Trigonometric Functions 19
6 Solving Trigonometric Equations 23
7 Applications with Right Triangles 29
8 Law of Sines and Law of Cosines 34
9 Area of a Triangle 41
10 Area of a Circular Sector 47
Blackline Masters 52
Solutions to the Activities 89

Graphs of Trigonometric Functions/TRIGONOMETRY USING THE SHARP EL-9600 1
GRAPHS OF
TRIGONOMETRIC FUNCTIONS
Chapter one
Introducing the Topic
In this chapter, you and your students will learn how to graph trigonometric
functions on the Sharp EL-9600. Graphing trigonometric functions on the
calculator is covered first, so that graphical evaluation of trigonometric
functions may be covered in the next chapter. The trigonometric functions
are often introduced as circular functions with angle measurement in degrees
and radians. The sine and cosine functions are said to map the angle
measurement to the yand xcoordinates respectively, of the point on the unit
circle representing the angle. Then, the four additional trigonometric functions
are defined in terms of sine and cosine.
If you let the angle measurement be represented by a horizontal axis (x) and let
the vertical axis (y) represent the function evaluated at the angle, then you can
graph the six trigonometric functions as y= sin x, y= cos x, y= tan x, y= csc x,
y= sec x, and y= cot x.

Calculator Operations
Turn the calculator on and press Y= . The Y prompts will appear in the viewing
window. Press CL to remove an old Y1 expression. Press ▼CL to clear
additional expressions. The calculator should be setup in radian measurement,
with rectangular coordinates, and in equation editor mode. To complete this
setup, press 2ndF SET UP , touch B DRG, double touch 2 Rad, touch
E COORD, double touch 1 Rect, touch G EDITOR, and double touch 1 Equation.
Press 2ndF QUIT again to exit the menu and return to the Y prompts.
To enter the sine function ( y= sin x) for Y1, press sin X/θ/T/n.
Enter the viewing window range by pressing ZOOM , touching E TRIG, and
double touching 1 sin x. This establishes built-in viewing-window range for
graphing the sine function. The window ranges are -2π≤x≤2πand -1.55 ≤y
≤1.55. The scale for the xaxis is and the y-axis scale is .5.
The remaining five trigonometric functions can be graphed similarly.
2Graphs of Trigonometric Functions/TRIGONOMETRY USING THE SHARP EL-9600
π
2

Method of Teaching
Use the Blackline Master 1.1 to create an overhead for graphing the sine
function. Go over in detail how to graph a trigonometric function. Follow
the sine function demonstration with the cosecant function demonstration
provided in the sample problem below and on the Blackline Master.
Next, use the Blackline Masters 1.2 through 1.4 to create worksheets for the
students. Have the students graph the remaining four trigonometric functions.
Use the topics For Discussion to supplement the worksheets. Engage the trace
feature by pressing TRACE . Move the tracer by pressing or . Tracing
will assist you in discussing the topics.
Using Blackline Master 1.1
The problem discussed above in the Calculator Operations section is presented
on Blackline Master 1.1. The following problem also appears on the
Blackline Master.
Graph the cosecant function by pressing Y= CL to remove an old Y1
expression. To enter the cosecant function for Y1, press MATH , you may
need to touch A CALC if the menu is not already there. Double touch 10 csc,
and press X/θ/T/n. Use the sine function's built-in viewing range, and press
GRAPH to view the graph.
Graphs of Trigonometric Functions/TRIGONOMETRY USING THE SHARP EL-9600 3
▼
▼

For Discussion
You and your students can use the calculator to discuss:
(1) intervals where the functions increase and decrease,
(2) domain and range of the functions,
(3) the periodicity of the functions, and
(4) which part of the horizontal axis represents each quadrant.
The trace feature will allow you to move along the curve to see where the
function is increasing and decreasing, to see the values associated with limits
on the domain and range, to see the period of a function, and to explore the
quadrant "markers" on the x-axis.
4Graphs of Trigonometric Functions/TRIGONOMETRY USING THE SHARP EL-9600

Introducing the Topic
In this chapter, you and your students will learn how to evaluate trigonometric
functions directly in radians or degrees, and graphically through the tracing feature.
Calculator Operations
To evaluate sin 45° directly, you will turn the calculator on and press to
enter computation mode. Press CL to clear the viewing window of any old
expressions. To evaluate trigonometric functions in degrees, you can set the
calculator to operate in degrees by pressing 2ndF SET UP , touch B DRG,
and double touch 1 Deg. Press 2ndF QUIT to exit the menu. Now, to find the
sin 45° you will press sin 4 5 ENTER .
Evaluating Trigonometric Functions/TRIGONOMETRY USING THE SHARP EL-9600 5
EVALUATING TRIGONOMETRIC
FUNCTIONS
Chapter two
×
+–
÷

To approximate sin 45° graphically, you will press Y= followed by CL to clear
an old expression for Y1. Enter the function sin xby pressing sin X/θ/T/n.
To view the graph press ZOOM , touch E TRIG, and double touch 1 sin x.
Now, engage the trace by pressing TRACE and moving the tracer near
x= 45° by pressing .
With the cursor located near x= 45°, you can zoom in to find a better
approximate for the yvalue. First, set the factors for zooming by pressing
ZOOM , touching B FACTOR, and pressing ENTER 5 ENTER 5 ENTER .
These larger values will accelerate the zooming in. Now, press ZOOM , touch
A ZOOM, and double touch 3 In to zoom in on the cursor. Once again, trace
over near x= 45°. Zoom in two more times to see a better approximation.
To find sin directly, you will turn the calculator on and press to enter
computation mode. Press CL to clear the viewing window of any old
expressions. To evaluate trigonometric functions in radian mode, you can set
the calculator to operate in radians by pressing 2ndF SET UP , touching
B DRG, and double touching 2 Rad. Press 2ndF QUIT to exit the menu.
Now, to find sin you will press sin 2ndF πa/b 3 ENTER .
6Evaluating Trigonometric Functions/TRIGONOMETRY USING THE SHARP EL-9600
π
3
π
3
×
+–
÷
▼

To approximate sin graphically, you will press Y= followed by CL to clear
an old expression for Y1. Enter the function sin xby pressing sin X/θ/T/n.
View the graph by pressing ZOOM , touching E TRIG, double touching 1 sin x.
Now, engage the trace by pressing TRACE and tracing near x= = 1.047.
With the cursor located near x= 1.047, you can zoom in to find a better
approximation for the yvalue. Press ZOOM , touch A ZOOM, and double
touch 3 In to zoom in on the cursor. Once again, trace over near x= 1.047.
Zoom in one more time to see an even better approximation.
Evaluating Trigonometric Functions/TRIGONOMETRY USING THE SHARP EL-9600 7
π
3
π
3

Method of Teaching
Use Blackline Master 2.1 to create an overhead or handout for direct and
graphical evaluation of trigonometric functions. Demonstrate the procedure
changes for radian measurement. Remind the students of the location of
cosecant, secant, and cotangent within the MATH menu.
Next, use the Blackline Masters 2.2 through 2.4 to create worksheets for the
students. Have the students evaluate the six trigonometric functions directly
and graphically for the various degree and radian measurements. Use the
topics For Discussion to supplement the worksheets.
Using Blackline Master 2.2
The following problem appears on Blackline Master 2.2. Evaluate sec directly
in degree mode. Press CL to enter computation mode and clear the home
screen. Press 2ndF SET UP to verify the calculator is in degree mode. If it is
not, change it to degree mode. Press 2ndF QUIT to exit the menu. Enter and
evaluate the expression by pressing MATH , touching A CALC, double touching
09 sec, pressing 2ndF πa/b 4 MATH , touching E ANGLE, double
touching 4 r, and pressing ENTER .
Evaluate tan 20° graphically. Press Y= and press CL to remove an old
expression for Y1. Press 2ndF SET UP to verify the calculator is in degree
mode. If it is not, change it. Press 2ndF QUIT to exit the menu. Enter the
function tan xfor Y1 by pressing tan X/θ/T/n. Instead of graphing the
function in the default viewing window for tangent, press WINDOW adjust the
Xmin and Xmax to 10° and 30° (value of interest ± 10° ) respectively to start out
closer to the value of interest. Set the Xscl = 1. Press ZOOM , touch A ZOOM,
8Evaluating Trigonometric Functions/TRIGONOMETRY USING THE SHARP EL-9600
π
4
▼
×
+–
÷

and double touch 1 Auto to view the graph. Now, engage the trace by pressing
TRACE . Notice the tracer appears at x= 20° since it is the middle of the
xvalues.
For Discussion
Students can calculate sin 25° in radian mode by pressing sin 2 5 MATH ,
touching E ANGLE, double touching 1°, and pressing ENTER . Students can
store a graph with a particular viewing window for repeated use by using the
StoGD command in the DRAW menu. For more detail, refer to the EL-9600
Owner's Manual. Discuss the use of the negative sign like cos (–10°). This is the
first assignment that uses a negative angle. Also remind them of the difference
between the negative key (a short dash within parentheses) and the minus key
(no parentheses).
Additional Problems
Evaluate the following directly:
(1) cos 27°
(2) cot 52°
(3) sec
Evaluate the following graphically:
(4) sin 107°
(5) tan
(6) csc
Evaluating Trigonometric Functions/TRIGONOMETRY USING THE SHARP EL-9600
5π
3
5π
4
π
7

Introducing the Topic
In this chapter, you and your students will learn how to verify trigonometric
formulas and identities graphically. The left and right sides of an equality can be
graphed by the calculator as Y1 and Y2. Verification is made by viewing Y1 and
Y2 as the same graph, hence equivalent expressions.
Calculator Operations
To verify the double angle formula sin 2θ= 2 sin θcos θ, you will graph the "sin
2θ" as Y1 = sin (2x) and the "2 sin θcos θ" as Y2 = 2 sin xcos xand show that
they are in fact the same graph. To do this you will turn the calculator on and
press Y= . Press CL to clear Y1 of an old expression. Press ▼CL to clear
additional expressions. Make sure the calculator is in radian mode by pressing
2ndF SET UP , touch B DRG, and double touch 2 Rad. Press 2ndF QUIT
to exit the menu.
Now enter sin (2x) for Y1 by pressing sin ( 2 X/θ/T/n)ENTER
and the 2 sin xcos xfor Y2 by pressing 2 sin X/θ/T/ncos X/θ/T/n
ENTER . Set your viewing window by pressing WINDOW EZ , touch 5, touch
on the screen, touch 6 –2π<X,<2π, touch on the screen, and touch 5 –5<Y<5.
10 Graphical Verification of Formulas and Identities/TRIGONOMETRY USING THE SHARP EL-9600
GRAPHICAL VERIFICATION OF
FORMULAS AND IDENTITIES
Chapter three
➧
➧

To verify the identity sin2θ+ cos2θ= 1, you will graph the "sin2θ+ cos2θ" as
Y1 = sin2x+ cos2xand the "1" as Y2 = 1, and show that they are the same
graph. To do this press Y= to access the Y1 prompt. Press CL to clear Y1 of
an old expression, and enter the sin2x+ cos2xfor Y1 by pressing ( sin
X/θ/T/n) ab2 + ( cos X/θ/T/n) ab2 ENTER . Enter the
1 for Y2 by pressing CL 1 ENTER . Press GRAPH to view the graph.
Method of Teaching
Use Blackline Master 3.1 to create an overhead or handout for graphically
verifying trigonometric formulas and identities. Demonstrate the process
of entering the equality's left- and right-hand expressions and when
necessary, emphasize the intervals where functions are positive or negative
(as demonstrated in Using Blackline Master 3.2). Next, use the Blackline Masters
3.3 through 3.5 to create worksheets for the students. Have the students verify
the various formulas and identities. Use the topics For Discussion to supplement
the worksheets.
Graphical Verification of Formulas and Identities/TRIGONOMETRY USING THE SHARP EL-9600 11
▼

Using Blackline Master 3.2
The following problem appears on Blackline Master 3.2. Verify the half angle
formula cos = ± (1 + cos ). To verify the formula, you will graph the “cos
” as Y1 = cos and the “± ” as Y2 = ± and show that they
are in fact the same graph. In this problem, you will need to limit your domain
or xvalues to intervals where cosine is either negative or positive and enter the
appropriate radical for Y2.
To do this, you will turn the calculator on and press Y= . Press CL
to clear Y1 of an old expression.
Now enter the cos for Y1 by pressing cos X/θ/T/na/b 2 ENTER and
the for Y2 (limiting to where cosine is positive) by pressing CL
2ndF a/b 1 + cos X/θ/T/n▼2 ENTER . Cos will be
positive for a viewing window of 0 < x< π. Enter this viewing window by
pressing WINDOW 0 ENTER 2ndF πENTER 2ndF π÷2 ENTER
(–) 3 ENTER 3 ENTER 1 ENTER . Press GRAPH to view the graphs.
Notice they appear as one graph.
To verify the formula for where cosine is negative, press Y= ▼to access the
Y2 prompt and press (–) to insert a negative in front of the radical. Cos will
be negative for a viewing window of π< x< 3π. Enter this viewing window by
pressing WINDOW 2ndF πENTER 3 2ndF πENTER .
Press GRAPH to view the graphs. Notice they appear as one graph.
12 Graphical Verification of Formulas and Identities/TRIGONOMETRY USING THE SHARP EL-9600
θ
2
θ
2
θ
21 + cos θ
2
1 + cos θ
2
x
2
x
2
x
2
x
2
1 + cos x
2
√
√√ √

For Discussion
Students can use the calculators to discuss how "toggling" (using the ▲and
▼keys while tracing) between Y1 and Y2 verifies the same values for both Y1
and Y2, and they can explore what happens when they do not limit the domain
on those problems with ± parts (they see two graphs).
Additional Problems
Verify the following formulas and identities graphically.
1. cos 2θ= cos2θ– sin2θ
2. 1 + cot2θ= csc2θ
3. sin2θcot2θ= (1 + sin θ)(1 – sin θ)
Graphical Verification of Formulas and Identities/TRIGONOMETRY USING THE SHARP EL-9600 13

Introduction
In this chapter, you and your students will learn how to modify trigonometric
functions by constants. If you multiply sine or cosine by a constant A, then
every value of the function is multiplied by A. The maximum absolute value of
the sine or cosine function is 1. Therefore, Asin θand Acos θwill have a
maximum value of |A|. The value of |A| is called the amplitude of the functions
y= Asin xand y= Acos x, and is the distance sine and cosine rises and falls
above the x-axis. Multiplication of the function by A will stretch or shrink the
curve vertically. |A| > 1 will stretch the curve while an |A| between 0 and 1
will shrink the curve. The tangent function does not have an amplitude because
tangent increases without bound as θapproaches values such as .
If we multiply the argument θby a constant kwithin a trigonometric function,
then every measure of an angle θis multiplied by k. Multiplication of θwill
stretch or shrink the curve horizontally. A value of k> 1 will shrink the curve
while a value of kbetween 0 and 1 will stretch the curve. The stretching and
shrinking of a curve horizontally effects the function's period. The new periods
will be for sine, cosine, secant and cosecant, and for tangent and cotangent.
14 Amplitude, Period, and Phase Shift/TRIGONOMETRY USING THE SHARP EL-9600
AMPLITUDE, PERIOD,
AND PHASE SHIFT
Chapter four
π
2
2π
kπ
k

If we add a constant cto the argument θwithin a trigonometric function, then
every measure of an angle θis increased by c. Addition of cwill shift the
function left or right. A c> 0 will shift the function to the left while a c< 0 will
shift the function to the right.
If we combine both the multiplication by a constant kand the addition of a
constant cto the argument θ(sin (kθ+ c)), then the movement of the function
left or right with respect to the period change is called the phase shift. The
value for this shift is . A c> 0 will shift the function units to the left while a
c< 0 will shift the function units to the right.
Calculator Operations
To investigate changes in amplitude, you will use the SHIFT/CHANGE feature of
the graphing calculator. Press 2ndF SHIFT/CHANGE , touch B CHANGE, and
double touch 5 Y=sin X. Double touch the screen at Y = 3, above the box with
the arrows. Notice the new function Y = 3 sin X is displayed. Press ENTER to
place the new function in the list.
Double touch the screen at Y = -2, below the box with the arrows. Notice the
new function Y = -2 sin X is displayed. Press ENTER to place the new function
in the list.
Amplitude, Period, and Phase Shift/TRIGONOMETRY USING THE SHARP EL-9600 15
-c
kc k
c
k

Touch the top of the right side of the screen and the list of three functions will
be displayed. The function Y = sin X is highlighted on the right side of the
screen, and the graph is darkened on the left side of the screen. Press ▼to
move from one graph to the next. Observe and note the changes in sin X when
you change the amplitude.
Method of Teaching
Use Blackline Master 4.1 to create an overhead or handout for demonstrating
changes in amplitude. Next use Blackline Masters 4.2 and 4.3 to create work-
sheets to demonstrate changes in period and phase shift. For analysis of period,
demonstrate the process of entering a "standard" trigonometric function for Y1
and a changed function for Y2. Use the trace feature to verify the new period.
For analysis of phase shift, demonstrate the use of SHIFT feature. Use the topics
For Discussion to supplement the worksheets.
Using Blackline Masters 4.2 and 4.3
The following problems appear on Blackline Masters 4.2 and 4.3. To investigate
the change in a trigonometric function by multiplying the argument θby 2, press
Y= CL to access and clear Y1 of an old expression. Make sure the calculator
is in radian mode by pressing 2ndF SET UP , touch B DRG, and double touch
2 Rad. Press 2ndF QUIT to exit the menu.
Now, enter sin xfor Y1 by pressing sin X/θ/T/nENTER and the sin (2x) for
Y2 by pressing CL sin ( 2 X/θ/T/n) ENTER . Set your viewing
window by pressing ZOOM , touching E TRIG, and double touching 1 sin x.
Note the change in the second graph versus the first graph. Engage the trace to
show the period has changed from 2πto π.
16 Amplitude, Period, and Phase Shift/TRIGONOMETRY USING THE SHARP EL-9600