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Sharp EL-9600, User Manual

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Sharp EL-9600 User Manual Download Page 23

Introducing the Topic

In this chapter, you and your students will learn how to compute and graph

inverse trigonometric functions.  The inverse is found by interchanging the

range and domain of the trigonometric function.  Thus, the domain of the

trigonometric function becomes the range of the inverse and the range of the

trigonometric function becomes the domain of the inverse.

For example, the sine function has IR as its domain and -1 

≤

y

≤

1 as its range.

Therefore, the inverse of the sine function (arcsine or inverse sine) will have 
-1 

≤

x

≤

1 as its domain and IR as its range.  Notice, like some algebraic functions,

the inverse of trigonometric function will not be a function.  This is due to 

periodicity of the trigonometric function's domain extends to the inverse's

range.  Thus, one value in the inverse's domain will correspond to more than

one value in the inverse's range.  To make the inverse a function, you will limit

the domain of the trigonometric function so it forms a one-to-one function.

Limit the domain of the sine and tangent functions to -

π

/2 

≤

x

≤ π

/2, and limit the

cosine function to 0 

≤

x

≤ π

.  These limited domains result in one-to-one functions

whose inverses are functions.  The values within these limited domains are called

the principal values.  Notations for "inverse sine" are "arcsin" and "sin

–1

".

Inverse Trigonometric Functions/TRIGONOMETRY USING THE SHARP EL-9600

19

INVERSE TRIGONOMETRIC
FUNCTIONS

Chapter five

Summary of Contents for EL-9600

Page 1: ...A P P L Y I N G U S I N G T H E EL 9600 D A V I D P L A W R E N C E TRIGONOMETRY SHARP Graphing Calculator ...

Page 2: ...cing 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 includ...

Page 3: ...y 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...

Page 4: ... 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 ...

Page 5: ...rigonometric 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 y and x coordinates 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 ...

Page 6: ...ch 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 2π and 1 55 y 1 55 The ...

Page 7: ...e 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 ...

Page 8: ... 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 4 Graphs of Trigonometric Functions TRIGONOMETRY USING THE SHARP EL 9600 ...

Page 9: ...tor 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...

Page 10: ...ER 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 i...

Page 11: ...in 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 y value 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 ...

Page 12: ...ode 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 π a 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...

Page 13: ...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 ...

Page 14: ...t 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 x cos x for Y2 by pressing 2 sin X θ T n...

Page 15: ...kline 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 t...

Page 16: ...miting 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 π ENTER 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 inse...

Page 17: ...nd 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 cos2 θ sin2 θ 2 1 cot2 θ csc2 θ 3 sin2 θ cot2 θ 1 sin θ 1 sin θ Graphical Verification of Formulas and Identities TRIGONOMETRY USING THE SHARP EL 9600 13 ...

Page 18: ...ch 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 θ approaches values such as If we multiply the argument θ by a constant k within a trigonometric function then every measure of an angle θ is multiplied by k Multiplication of θ will stretch or shrink the curve horizontally A value of k 1 will shr...

Page 19: ...his 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 ...

Page 20: ... 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 θ by 2 press Y CL to access and clear Y1 of an old expression Make sure ...

Page 21: ...n at Y 2 below the box with the arrows Notice the new function Y sin X 3π 2 2 is displayed Press ENTER to place the new function in the list 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 ne...

Page 22: ...gent and what happens when you multiply tangent by a constant Additional Problems Use the following to demonstrate the period period changes and phase shifts for tangent 1 Compare tan θ with 2 tan θ 2 Compare tan θ with tan 3 Compare tan θ with tan θ 4 Compare tan θ with tan θ 2 18 Amplitude Period and Phase Shift TRIGONOMETRY USING THE SHARP EL 9600 θ 3 π 2 π 2 ...

Page 23: ... the inverse of trigonometric function will not be a function This is due to periodicity of the trigonometric function s domain extends to the inverse s range Thus one value in the inverse s domain will correspond to more than one value in the inverse s range To make the inverse a function you will limit the domain of the trigonometric function so it forms a one to one function Limit the domain of...

Page 24: ...u will turn the calculator on and press CL to enter computation mode and clear the home screen The computations can be done in degrees and radians Let s do the first one in degrees by pressing 2ndF SET UP touching B DRG and double touching 1 Deg Press 2ndF QUIT to exit the menu To find sin 1 5 press 2ndF sin 1 5 ENTER The answer of 30 is shown below To graph the inverse sine function press Y CL to...

Page 25: ...sheets Using Blackline Master 5 2 The following problems appear on Blackline Master 5 2 To compute inverse trigonometric functions you will turn the calculator on and press CL to enter computation mode The computations can be done in degrees and radians Let s do this one in radians by pressing 2ndF SET UP touching B DRG double touching 2 Rad Press 2ndF QUIT to exit the menu To find sin 1 4 press 2...

Page 26: ...f a trigonometric function with its inverse Demonstrate to the students the location of inverse secant inverse cosecant and inverse cotangent within the MATH menu Additional Problems Compute the following 1 sin 1 7 _______________ degrees 2 cos 1 5 _______________ degrees 3 tan 1 2 _______________ degrees 4 csc 1 2 _______________ degrees 5 sin 1 2 _______________ radians 6 cos 1 6 _______________...

Page 27: ...on as Y1 and the right side as Y2 In the case where the right side of the equation is zero you will graph the left side as Y1 and solve for the intercepts The point s of intersection will be the solutions to the equation Calculator Operations To solve a trigonometric equation graphically you will turn the calculator on and press Y CL to access and clear Y1 of an old expression Solving can be done ...

Page 28: ...g ZOOM touching B FACTOR pressing ENTER 5 ENTER 5 ENTER Press GRAPH to exit the menu and return to the graph These larger factors will expedite the zooming in or zooming out that you desire to do To find the left intercept press TRACE to engage the trace and move the cursor near this intercept by continuing to press the key Your graph should look like the following one Now press ZOOM touch A ZOOM ...

Page 29: ...cally Demonstrate both the solving by zooming described above in Calculator Operations and the CALC feature demonstrated in Using Blackline Master 6 2 In addition solve equations in both radian and degree modes Next use the Blackline Master 6 3 to create a worksheet for the students Have the students find the solutions to the trigonometric equations through zooming and calculation in both radian a...

Page 30: ...o find the solutions of 2 sin2 x cos x 1 0 press 2 sin X θ T n ab 2 cos X θ T n 1 ENTER Set your viewing window to the principal values for sine and cosine by pressing WINDOW 2ndF π 2 ENTER 2ndF π ENTER 2ndF π 4 ENTER 1 ENTER 1 ENTER 1 ENTER Press GRAPH to view the graph To calculate an intercept solution press 2ndF CALC and double touch 5 X_Incpt You can calculate another intercept by pressing 2n...

Page 31: ... 1 ENTER Set your viewing window to the combined principal values by pressing WINDOW 2ndF π 2 ENTER 2ndF π ENTER 2ndF π 4 ENTER 2 ENTER 2 ENTER 1 ENTER Press GRAPH to view the graph There appears to be four solutions to the equation To calculate the intersections solutions press TRACE move the tracer right until it appears on the screen and press 2ndF CALC and double touch 2 Intsct You can calcula...

Page 32: ...o verify the additional solutions at integral multiples of the function s period Additional Problems Solve the following equations using either the zoom or CALC feature 1 2cos2 x 5cos x 2 0 in degrees 2 tan 2 x cot x in degrees 3 2cos2 x sin x 1 in degrees 4 3tan2 x 4sec x 4 0 in radians 5 cos x tan x sin2 x 0 in radians 6 2sin x cos x 4sin x cos x 2 in radians 28 Solving Trigonometric Equations T...

Page 33: ...or Operations Before using the calculator to solve problems in this chapter set the calculator to operate in degrees by pressing 2ndF SET UP touching B DRG and double touching 1 Deg Press 2ndF QUIT to exit the set up menu Historically the six trigonometric functions were defined in relation to a right triangle sin θ csc θ cos θ sec θ tan θ cot θ Applications with Right Triangles TRIGONOMETRY USING...

Page 34: ...bles Press to move the blinking cursor over the A and press 3 ENTER Type in the value of C by pressing 6 ENTER Solve for θ by pressing 2ndF EXE To use the SOLVER to find B edit the current equation to the new equation cos θ Do this by pressing CL CL to display the current equation sin θ With the cursor over the s in sin press cos to insert cos Press DEL to delete sin Press to move the cursor over ...

Page 35: ... the topics For Discussion to supplement the worksheets While in the SOLVER press 2ndF SOLVER touch A METHOD and notice that the SOLVER utilizes three methods of solving equations The Equation method is discussed in this chapter The Graphic and Newton methods will be discussed in Chapter 8 Using Blackline Master 7 2 A rectangle has length 16 cm and width 8 6 cm Find the degree measure of angle A D...

Page 36: ...d θ To determine the length of the diagonal of the rectangle return to the equation by pressing CL CL and edit the current equation to read sin θ The value of θ should be as before so enter 8 6 for A and solve for C For Discussion Students can edit a current equation in the SOLVER to a new equation instead of clearing the current equation and typing in the new equation 32 Applications with Right T...

Page 37: ...ween points C and D to be 4 if S 2 and R 3 5 Explain why or why not 6 A farmer wants to put a fence around the perimeter of a flat field that is in the shape of a right triangle One angle of the triangle is 42 and the length of the side opposite this angle is 60 meters Find the length of fencing that is required to enclose the field Applications with Right Triangles TRIGONOMETRY USING THE SHARP EL...

Page 38: ... solve problems in this chapter set the calculator to operate in degrees by pressing 2ndF SET UP touching B DRG and double touching 1 Deg Press 2ndF QUIT to exit the set up screen The Law of Sines states that the ratio of the lengths of the sides of a triangle to each other is the same as the ratio of the sines of the angles opposite the sides That is if A B and C are the measures of the three ang...

Page 39: ...emory by pressing 2ndF SOLVER touching C SAVE pressing ENTER and typing in a name of your choice say LAWSINES by pressing L A W S I N E S Note that the cursor is automatically set to ALPHA so you can just type letters Press ENTER to store the formula The equation you have stored will reappear on the screen Press 2ndF SOLVER touch A METHOD double touch 1 Equation to choose the Equation Solver Press...

Page 40: ... first part of this example angle A measures 95 side x is of length 15 and angle B measures 48 To start the problem from the beginning enter 0 for y Use to position the cursor over the unknown quantity y To begin the solution process press 2ndF EXE and you will see You must enter the range values for the unknown from BEGIN to END within which you expect to find the solution Let the BEGIN 5 by pres...

Page 41: ... solutions In this case however you have included from BEGIN to END all possible values for the measure of angle B Thus there is no triangle that could have the indicated measures You can algebraically confirm this fact by noticing that sin B Y sin A X 3 sin 65 2 719 1 which is of course impossible The Law of Cosines is helpful in solving problems when the lengths of two sides and the angle betwee...

Page 42: ...ENTER to view the list of variables Z X Y and θ Press to position the cursor on X and enter 13 Next enter 8 for Y and 50 for θ Press to move the cursor over Z and press 0 ENTER Press 2ndF EXE to solve for Z Notice that the Equation Solver has automatically changed to the Newton Solver The calculator has chosen this method of solution since the equation is such that a simple rearrangement of the va...

Page 43: ...e Law of Sines Use Blackline Master 8 2 to create an overhead or handout for using the Newton Solver with the Law of Cosines Demonstrate the procedure to store each of these equations in the calculator memory for later recall Next use Blackline Master 8 3 to create a worksheet for the students Remind them that some problems may be solved by a variety of methods If information is not available to o...

Page 44: ...5 76 15 yielding θ 13 85 For Discussion Students can discuss how to retrieve and rename the stored equations for the Law of Sines and the Law of Cosines Additional Problems Problems 1 3 refer to the following situation A tower on the top of a hill is 40 feet tall The angle of elevation from point B at the base of the hill to the top of the tower point T is 45 The angle of elevation from point A 80...

Page 45: ...DRG and double touch 1 Deg Press 2ndF QUIT to exit the set up screen Depending on the information given in the problem situation there are three formulas you can use to find the area of any triangle 1 The area of a triangle is one half the product of the base and the height where the height is the length of the line segment drawn from a vertex of the triangle perpendicular to the opposite side cal...

Page 46: ...Z The first formula for the area of a triangle area base height can be typed as A B H Note that the multiplication key must be pressed between the base and height to separate the variable names To access the SOLVER press 2ndF SOLVER and press CL to delete any equation on the screen Type in the equation A B H with the keystrokes ALPHA A ALPHA a b 1 2 ALPHA B ALPHA H Press 2ndF SOLVER touch A METHOD...

Page 47: ...escribed in Chapter 7 The second formula for the area of a triangle can be typed in as A B C sin θ Press CL three times to return to and delete the current equation Type in the equation A B C sin θ with the keystrokes ALPHA A ALPHA a b 1 2 ALPHA B ALPHA C sin ALPHA θ Consider the triangle below where sides of lengths 4 8 and 5 73 enclose a 47 angle The height of this triangle is not known but you ...

Page 48: ...Discussion to supplement the worksheets Using Blackline Master 9 2 The second problem presented under the CALCULATOR OPERATIONS section also provides a nice exercise with the Graphic solver Press 2ndF SOLVER touch A METHOD and double touch 3 Graphic to choose the Graphic solver To start the problem from the beginning set A to 0 Move the cursor to the A variable and press 2ndF EXE to see the Graphi...

Page 49: ... Z Note that the square root symbol extends over the entire right hand side of the equation automatically grouping the expression under it You could enter the semiperimeter S X Y Z in a separate formula in the SOLVER but it is probably easier to find the value of S in the computation mode and then use Heron s formula in the SOLVER Press ENTER to view the variable list For Discussion Students can d...

Page 50: ...find the area of triangle ACB 4 Find the area of a triangle with vertices at 1 5 1 1 and 4 2 5 Suppose you are told that a triangle with two sides of length 12 and 15 has an area equal to 200 and you are asked to find the measure of the angle between the two given sides What would you find Why did you find it 46 Area of a Triangle TRIGONOMETRY USING THE SHARP EL 9600 A D C a b c d h α β υ θ B ...

Page 51: ...calculator to operate in radians by pressing 2ndF SET UP touching B DRG and double touching 2 Rad Press 2ndF QUIT to exit the set up screen A portion of a curve is called an arc When the curve is a circle a central angle is an angle whose vertex is at the center of the circle The central angle is said to intercept or subtend the arc Area of a Circular Sector TRIGONOMETRY USING THE SHARP EL 9600 47...

Page 52: ...s the SOLVER by pressing 2ndF SOLVER Press CL to delete any equation that appears on the screen Type in the equation A R2 θ by pressing ALPHA A ALPHA 1 a b 2 ALPHA R x2 ALPHA θ Consider this problem situation A lawn sprinkler is set so that the region of the lawn that is watered is approximately a sector of a circle of radius 6 feet If the central angle of the sector is radians find the area of th...

Page 53: ...ing the Newton solver first press 2ndF SOLVER touch A METHOD and double touch 2 Newton Reset A to 0 to start the problem from the beginning The values of R and θ are correct for this problem so use to place the cursor over the A variable and press 2ndF EXE to view the Newton solver screen Since area is never negative you could set START 0 Leave the STEP size as 001 and remember that you should cha...

Page 54: ...techniques Equation Newton and Graphic available in the SOLVER so they will have experience with other methods if the one they are attempting to use fails Use the topics For Discussion provided to supplement the worksheets Using Blackline Master 10 2 To find the area using the Graphic Solver press 2ndF SOLVER touch A METHOD and double touch 3 Graphic Reset A to 0 to start the problem from the begi...

Page 55: ... whose arc subtends an angle of measure radians 2 A circular garden has diameter 8 feet Broccoli is to be planted in a sector of this garden whose arc subtends an angle of 50 Find the area of the broccoli sector 3 If the area of a sector of a circle is 7 52 square inches and the radius of the circle is 2 4 inches find the measure of the angle subtended by the arc of the sector Give units with your...

Page 56: ...MASTERS 3 1 3 5 61 65 AMPLITUDE PERIOD AND PHASE SHIFT BLACKLINE MASTERS 4 1 4 3 66 68 INVERSE TRIGONOMETRIC FUNCTIONS BLACKLINE MASTERS 5 1 5 3 68 71 SOLVING TRIGONOMETRIC EQUATIONS BLACKLINE MASTERS 6 1 6 3 72 74 APPLICATIONS WITH RIGHT TRIANGLES BLACKLINE MASTERS 7 1 7 3 75 77 LAW OF SINES AND LAW OF COSINES BLACKLINE MASTERS 8 1 8 3 78 80 AREA OF A TRIANGLE BLACKLINE MASTERS 9 1 9 4 81 84 AREA...

Page 57: ...nd double touch 1 Equation 5 Press 2ndF QUIT again to return to the Y prompts 6 Enter the sine function y sin x for Y1 by pressing sin X θ T n 7 Enter the viewing window range by pressing ZOOM touching E TRIG and double touching 1 sin x Steps for graphing the cosecant trigonometric function 1 Press Y CL to remove an old Y1 expression 2 Enter the cosecant function for Y1 by pressing MATH touch A CA...

Page 58: ... To set up the calculator 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 5 Press 2ndF QUIT again to return to the Y prompts 6 Enter the cosine function y cos x for Y1 by pressing cos X θ T n 7 Enter the viewing window range by pressing ZOOM touching E TRIG and double touching 2 cos x 8 Sketch your graph in the box GRAPH...

Page 59: ...4 To set up the calculator 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 5 Press 2ndF QUIT again to return to the Y prompts 6 Enter the cosine function y tan x for Y1 by pressing tan X θ T n 7 Enter the viewing window range by pressing ZOOM touching E TRIG and double touching 3 tan x 8 Sketch your graph in the box GRAP...

Page 60: ...CALC double touch 09 sec and press X θ T n 3 Use the cosine function s built in viewing range by pressing ZOOM touching E TRIG and double touching 2 cos x Activity 4 Graph the cotangent function Try graphing the cotangent function without looking at the steps provided in Activities 1 3 Use the built in viewing window range for tangent and sketch your graph in the box below GRAPHS OF TRIGONOMETRIC ...

Page 61: ...ollowed by CL to clear an old expression for Y1 2 Enter the function sin x by pressing sin X θ T n Set the calculator to radian mode by pressing 2nd F SET UP touch B DRG and double touch 2 Rad Press 2nd F QUIT to exit the menu 3 View the graph by pressing ZOOM touching E TRIG double touching 1 sin x 4 Now engage the trace by pressing TRACE and tracing near x 1 047 5 With the cursor located near x ...

Page 62: ...0 graphically 1 Press Y and press CL to remove an old expression for Y1 2 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 3 Enter the function tan x for Y1 by pressing tan X θ T n 4 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 re...

Page 63: ... Direct evaluation of trigonometric functions in degrees 1 sin 80 2 cos 10 3 tan 70 4 csc 55 5 sec 22 6 cot 40 Activity 2 Direct evaluation of trigonometric functions in radians 7 sin 8 cos 9 tan 10 csc 11 sec 12 cot EVALUATING TRIGONOMETRIC FUNCTIONS 2 3 NAME _____________________________________________________ CLASS __________ DATE __________ π 5 5π 4 2π 3 7π 6 π 4 π 7 ...

Page 64: ...proximation of trigonometric functions in degrees 13 sin 18 14 cos 33 15 tan 134 16 csc 85 17 sec 145 18 cot 235 Activity 4 Graphical approximation of trigonometric functions in radians 19 sin 20 cos 21 tan 22 csc 23 sec 24 cot EVALUATING TRIGONOMETRIC FUNCTIONS 2 4 NAME _____________________________________________________ CLASS __________ DATE __________ 4π 5 π 9 3π 8 6π 7 3π 4 5π 11 ...

Page 65: ...n Press CL to clear additional expressions 3 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 4 Now enter sin 2x for Y1 by pressing sin 2 X θ T n ENTER and the 2 sin x cos x for Y2 by pressing 2 sin X θ T n cos X θ T n ENTER 5 Set your viewing window by pressing WINDOW EZ touch 5 touch on the screen touch 6 2π X ...

Page 66: ...NTER and the for Y2 limiting to where cosine is positive by pressing CL 2ndF a b 1 cos X θ T n 2 ENTER 5 Cos will be positive for a viewing window of 0 x π Enter this viewing window by pressing WINDOW 0 ENTER 2ndF π ENTER 2ndF π 2 ENTER 3 ENTER 3 ENTER 1 ENTER 6 Press GRAPH to view the graphs Notice they appear as one graph 7 To verify the formula for where cosine is negative press Y to access the...

Page 67: ... 2ndF QUIT to exit the set up menu 3 Enter the sin for Y1 by pressing sin X θ T n a b 2 ENTER 4 Enter the for Y2 limiting domain to an interval where sine is positive by pressing 2ndF a b 1 cos X θ T n 2 ENTER 5 Sin is positive for 0 x 2π Enter this range by pressing WINDOW 0 ENTER 2 2ndF π ENTER 2ndF π 2 ENTER 2 ENTER 2 ENTER 1 ENTER 6 Check to see if they result in the same graph by pressing GRA...

Page 68: ...ive 1 Press Y to access the Y2 prompt 2 Press to insert a negative in front of the radical 3 Sin will be negative for 2π x 4π Enter this range by pressing WINDOW 2 2ndF π ENTER 4 2ndF π ENTER 4 Check to see if they result in the same graph by pressing GRAPH 5 Sketch the graph in the box below GRAPHICAL VERIFICATION OF FORMULAS AND IDENTITIES 3 4 NAME _______________________________________________...

Page 69: ...n2 x 1 for Y1 3 Enter the sec2 x for Y2 4 Set your viewing window 5 Check to see if they result in the same graph 6 Sketch the graph in the box below Activity 4 Verify the following formulas and identities using the steps provided in Activities 1 through 3 1 cos 2θ 2cos2 θ 1 2 csc2 θ cot2 θ 1 3 tan θ csc θ sec θ GRAPHICAL VERIFICATION OF FORMULAS AND IDENTITIES 3 5 NAME ___________________________...

Page 70: ...function Y 2 sin X is displayed 5 Press ENTER to place the new function in the list 6 Touch the top of the right side of the screen and the list of three functions will be displayed 7 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 8 Press to move from one graph to the next Observe and note the changes in sin X when you c...

Page 71: ... 2π to π Investigate the change in a trigonometric function by subtracting to the argument θ 1 Press 2ndF SHIFT CHANGE touch A SHIFT and double touch 6 Y sin X Double touch the screen at X 3π 2 to the right of the box with the arrows Notice the new function Y sin X 3π 2 is displayed Press ENTER to place the new function in the list 2 Double touch the screen at Y 2 below the box with the arrows Not...

Page 72: ... 2x ____________________________ 4 cos x with cos ____________________________ 5 cos x with cos x ____________________________ 6 cos x with cos x ____________________________ 7 cos x with 3cos x ____________________________ 8 tan x with 2tan ____________________________ 9 cot x with cot ____________________________ 10 sec x with 2sec 3x ____________________________ 11 csc x with csc x ____________...

Page 73: ...touching 1 Deg Press 2ndF QUIT to exit the menu 3 Find sin 1 5 by pressing 2ndF sin 1 5 ENTER The answer of 30 is shown below To graph the inverse sine function in degrees 1 Press Y CL to access and clear the Y1 expression 2 Press ENTER CL to clear additional Y expressions 3 Enter sin 1 x for Y1 by pressing 2ndF sin 1 X θ T n ENTER 4 Set your viewing window by pressing ZOOM touching E TRIG and dou...

Page 74: ...e touching 2 Rad Press 2ndF QUIT to exit the menu 3 Find sin 1 4 press 2ndF sin 1 4 ENTER The answer of 4115 radians is shown below To find the result in terms of π press 2ndF π ENTER To graph the inverse cosine function in radians 1 Press Y CL to access and clear the Y1 prompt 2 Enter cos 1 x for Y1 by pressing 2ndF cos 1 X θ T n ENTER 3 Set your viewing window by pressing ZOOM touching E TRIG an...

Page 75: ...ollowing 1 sin 1 8 _______________ degrees 2 cos 1 6 _______________ degrees 3 tan 1 2 _______________ degrees 4 sec 1 1 3 _______________ radians 5 csc 1 1 7 _______________ radians 6 cot 1 1 _______________ radians Graph the following 7 y tan 1 x 8 y cot 1 x INVERSE TRIGONOMETRIC FUNCTIONS 5 3 NAME _____________________________________________________ CLASS __________ DATE __________ ...

Page 76: ...9 0 ENTER 1 0 ENTER 1 ENTER 1 ENTER 1 ENTER Press GRAPH to view the graph 4 To save this original graph and viewing window press ZOOM touch G STO and double touch 1 StoWin 5 Set the zoom factors by pressing ZOOM touching B FACTOR pressing ENTER 5 ENTER 5 ENTER Press GRAPH to exit the menu 6 To find the left intercept press TRACE and move the cursor near this intercept by continuing to press the ke...

Page 77: ...alculate the other two intercepts by pressing 2ndF CALC and double touching 5 X_Incpt again To solve an equation with functions on both the left and right sides 1 Press Y CL to access and clear Y1 of an old expression To find the solutions of 2tan x sin x 2sin x tan x 1 you will enter the left side as Y1 by pressing 2 tan X θ T n sin X θ T n 2 sin X θ T n ENTER 2 Enter the right side as Y2 by pres...

Page 78: ...s2 x 1 _______________ degrees 4 sin 2x cos x _______________ radians 5 tan x cot x 2 _______________ radians Activity 2 Solve the following with the CALC feature 1 sin x cos x 0 _______________ degrees 2 sin2 x cos x 1 _______________ degrees 3 3cos 2x 5cos x 1 _______________ radians 4 3cot x sin x 2cos2 x 0 _______________ radians 5 4tan x sin 2x 0 _______________ radians SOLVING TRIGONOMETRIC ...

Page 79: ...A METHOD and double touch 1 Equation to choose the Equation solver Press ENTER and see the list of variables 5 Press to move the blinking cursor over the A and press 3 ENTER Type in the value of C by pressing 6 ENTER Solve for θ by pressing 2nd EXE 6 To use the SOLVER to find B edit the current equation to the new equation cos θ Do this by pressing CL CL to display the current equation sin θ With ...

Page 80: ... θ ALPHA a b ALPHA A ALPHA B 3 Press ENTER to view the variable list 4 Enter 8 6 for A and 16 for B The value of θ will be listed as the one that was used in the previously entered equation 5 Position the cursor over θ and press 2ndF EXE to find θ 6 To determine the length of the diagonal of the rectangle return to the equation by pressing CL CL and edit the current equation to read sin θ A C 7 Pr...

Page 81: ...he ramp What is the degree measure of the angle at the end of the ramp and level ground 3 A guy wire for a television antenna makes an angle of 60 with the mast If the wire is attached to the antenna 50 feet from the base of the antenna use the secant function to determine the length of the wire 4 John is lying on the beach 500 feet from his hotel and looking at Paula who is standing on a balcony ...

Page 82: ... 5 With the blinking cursor over the A type in 95 and press ENTER Next type 15 for x the length of the side opposite the 95 and press ENTER Enter 48 for B in the same manner Solve for y by positioning the blinking cursor over the y and press 2ndF EXE 6 You must enter the range values for the unknown from BEGIN to END within which you expect to find the solution Let the BEGIN 5 by pressing 5 ENTER ...

Page 83: ...lengths 13 cm and 8 cm enclose a 50 angle Find the length of the third side 5 Press ENTER to view the list of variables Press to position the cursor on X and enter 13 Next enter 8 for Y and 50 for θ Press to move the cursor over Z and press 0 ENTER 6 Press 2ndF EXE to solve for Z Notice that the SOLVER has automatically changed to the Newton Solver You should supply an initial estimate of the solu...

Page 84: ...Law of Cosines 2 A bridge is to be built between points B and C across a lake A surveyor using a transit determines the measure of the angle at point A to be 82 The distance between points A and B and between points A and C is known to be 125 yards Find the distance across the lake that the bridge will span 3 Two adjacent sides of a parallelogram are of length 8 and 12 The length of the shorter di...

Page 85: ...t with base 4 meters and height 8 35 meters use to move the blinking cursor over B and type in 4 Type in 8 35 for H and press ENTER Press twice to move the prompt to the A variable Press 2ndF EXE to find the area 16 7 square meters 6 Save the formula by pressing 2ndF SOLVER and touching C SAVE Enter a name and press ENTER 7 Enter A B C sin θ by pressing CL a couple times to return to and delete th...

Page 86: ...END is too small increase it and try again 4 Press 2ndF EXE and an autoscaled graph of the two sides of the equation is drawn and the solution is obtained 5 Enter Heron s formula A S S X S Y S Z press CL and type the following ALPHA A ALPHA 2ndF ALPHA S ALPHA S ALPHA X ALPHA S ALPHA Y ALPHA S ALPHA Z 6 Note that the square root symbol extends over the entire right hand side of the equation automat...

Page 87: ...epartment plans to purchase the region shown as MOR for frontage on a proposed interstate highway A survey of the region shows the distance from point M to point O is 60 feet the distance from R to M is 85 feet and the angle enclosed by these two lengths is 65 The deed for the lot indicates that the distance from S to T is 170 feet and the distance from S to R is 245 feet The owner of the lot has ...

Page 88: ...losed by the web Find the perimeter of the web 5 Yum Yum Burgers Corporation is considering purchasing a corner lot in town for a drive through hamburger restaurant They have decided to by the land if the price is 9000 or less If the lot is triangular in shape with sides of lengths 100 feet 50 feet 72 feet and the owner is asking 5 per square foot will Yum Yum Burgers purchase the lot 6 Jim is los...

Page 89: ...ver by pressing 2ndF SOLVER touch A METHOD and double touch 1 Equation Press ENTER and see the list of variables 5 Use to place the cursor over R type in 6 and press ENTER Type in θ by pressing 2ndF π 5 ENTER Press twice to place the cursor on the A variable Press 2ndF EXE to find the area in square feet 6 Use the Newton solver by pressing 2ndF SOLVER touch A METHOD and double touch 2 Newton Reset...

Page 90: ...and double touch 3 Graphic 3 Reset A to 0 to start the problem from the beginning 4 Place the cursor on the A variable and press 2ndF EXE 5 Recall that BEGIN and END are the lower and upper values between which you feel the solution will lie Enter a 20 for END so that the intersection of the two graphs is visible on the screen 6 Press 2ndF EXE to find the area AREA OF A CIRCULAR SECTOR 10 2 NAME _...

Page 91: ...zza in the shape of a circle has a diameter of 20 inches The pizza is cut into 8 slices of equal area What is the area of each piece of pizza Use the Graphic solver to find the area of a circular sector 4 A circular clock has the tip of the hour hand touching the rim of the clock What is the area of the sector formed when the hour hand moves from 1 o clock to 5 o clock 5 A cherry cheesecake is in ...

Page 92: ... CLIP PRGM DRAW INS RCL SET UP OPTION ex x 1 10x F P O N 9 w M 8 v L 7 u Q 4 L4 R 5 L5 S 6 L6 X 3 W 2 L3 L2 V 1 L1 O 0 T LIST SPACE ENTRY ANS ENTER Y e i Z EXE U FINANCE E K D J π C QUIT In log VARS X θ T n tan ab tan 1 a B cos cos 1 H I a b A sin sin 1 G Exp x2 A LOCK TOOL STAT ALPHA MATH 2nd F MATRIX OFF ON STATPLOT SOLVER Y DEL STO BS CL EZ GRAPH TABLE SUB TRACE ZOOM WINDOW SHIFT CHANGE SLIDE S...

Page 93: ...OD AND PHASE SHIFT ADDITIONAL PROBLEMS 1 flipped over rising faster 2 period is 3π 3 shifted right 4 shifted left and up 2 BLACKLINE MASTER 4 1 3 sin x has an amplitude of 3 2sin x is flipped over with an amplitude of 2 BLACKLINE MASTER 4 2 The function has moved 3 to the right and down 2 BLACKLINE MASTER 4 3 1 larger amplitude 2 smaller amplitude 3 smaller period 4 larger period 5 moved right 6 m...

Page 94: ... BLACKLINE MASTER 7 2 5 28 258 7 18 165 BLACKLINE MASTER 7 3 1 22 993 29 458 2 12 369 14 036 3 100 feet 4 41 987 8 LAW OF SINES AND LAW OF COSINES ADDITIONAL PROBLEMS 1 1447 761 2 766 986 3 502 341 4 1521 317 BLACKLINE MASTER 8 1 7 11 1898 BLACKLINE MASTER 8 2 7 9 9649 BLACKLINE MASTER 8 3 1 13 854 2 200 yards 3 59 395 9 AREA OF A TRIANGLE ADDITIONAL PROBLEMS 1 30 311 2 1 5 3 190 84 4 12 5 no inte...

Page 95: ...Teaching Notes TRIGONOMETRY USING THE SHARP EL 9600 91 TEACHING NOTES ...

Page 96: ...92 Teaching Notes TRIGONOMETRY USING THE SHARP EL 9600 TEACHING NOTES ...

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