Trigonometry: New Applications
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en-us2014 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemMon, 22 Dec 2014 04:17:08 GMTMon, 22 Dec 2014 04:17:08 GMTNew applications in the Trigonometry categoryhttp://www.mapleprimes.com/images/mapleapps.gifTrigonometry: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=155
Why I Needed Maple to Make Cream Cheese Frosting
http://www.maplesoft.com/applications/view.aspx?SID=125069&ref=Feed
<p>A recipe for cream cheese frosting I was making called for 8 oz. (about 240 grams) of cream cheese. Unfortunately, I didn't have a kitchen scale, and the product I bought came in a 400 gram tub in the shape of a<strong> truncated cone</strong>, which has a rather cumbersome volume formula. <br />Given the geometry of this tub, how deep into the tub should I scoop to get 240 grams? The mathematics is trickier than you might think but lots of fun! And the final, tasty result is worth the effort!</p><img src="/view.aspx?si=125069/philly_thumb.png" alt="Why I Needed Maple to Make Cream Cheese Frosting" align="left"/><p>A recipe for cream cheese frosting I was making called for 8 oz. (about 240 grams) of cream cheese. Unfortunately, I didn't have a kitchen scale, and the product I bought came in a 400 gram tub in the shape of a<strong> truncated cone</strong>, which has a rather cumbersome volume formula. <br />Given the geometry of this tub, how deep into the tub should I scoop to get 240 grams? The mathematics is trickier than you might think but lots of fun! And the final, tasty result is worth the effort!</p>125069Tue, 23 Aug 2011 04:00:00 ZDr. Jason SchattmanDr. Jason SchattmanClassroom Tips and Techniques: Yet More Gems from the Little Red Book of Maple Magic
http://www.maplesoft.com/applications/view.aspx?SID=102692&ref=Feed
Five more bits of accumulated "Maple magic" are shared: the limit of Picard iterates, combining radicals, factoring, yet another trig identity, and sorting strategies.<img src="/view.aspx?si=102692/thumb.jpg" alt="Classroom Tips and Techniques: Yet More Gems from the Little Red Book of Maple Magic" align="left"/>Five more bits of accumulated "Maple magic" are shared: the limit of Picard iterates, combining radicals, factoring, yet another trig identity, and sorting strategies.102692Mon, 21 Mar 2011 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Gems from the Little Red Book of Maple Magic
http://www.maplesoft.com/applications/view.aspx?SID=100897&ref=Feed
Five bits of "Maple magic" accumulated in recent months are shared: converting the half-angle trig formulas to radicals, tickmarks along a parametric curve, writing unevaluated math on a graph, changing Maple's differentiation formulas, and drawing a decent surface for a function containing a square root.<img src="/view.aspx?si=100897/thumb.jpg" alt="Classroom Tips and Techniques: Gems from the Little Red Book of Maple Magic" align="left"/>Five bits of "Maple magic" accumulated in recent months are shared: converting the half-angle trig formulas to radicals, tickmarks along a parametric curve, writing unevaluated math on a graph, changing Maple's differentiation formulas, and drawing a decent surface for a function containing a square root.100897Fri, 14 Jan 2011 05:00:00 ZDr. Robert LopezDr. Robert LopezTerminator circle with animation
http://www.maplesoft.com/applications/view.aspx?SID=100509&ref=Feed
<p>The idea of writing this article came to me on the 25th of June 2003 when I was listening to Cairo radio announcing that Maghrib prayer is due in Cairo city while I was sitting in my home town at 400 miles North East of Cairo. What is interesting is that at exactly the same time a next door mosque, in my home town, was also calling for the Maghrib prayer. This makes me wonder : how could it be that sunset is simultaneous at two locations separated by a distance of 400 miles from each other and at different Latitudes & Longitudes. As a reminder Maghrib prayer time occurs everywhere at sunset. <br />In what follows we explore this issue and try to prove or disprove the simultaneity of sunset at two different locations. In so doing we are led to some interesting conclusions and as a bonus we got ourselves an animation of the Terminator circle on the surface of the globe. <br />Aside from its modest value and its originality ( I am not aware of anything similar to it ) this article is a good exercise in Maple programming. <br />May this article be a stimulus for some readers to get interested in Astronomy which is a science as ancient as the early human civilizations. <br /><br /></p><img src="/view.aspx?si=100509/thumb.jpg" alt="Terminator circle with animation" align="left"/><p>The idea of writing this article came to me on the 25th of June 2003 when I was listening to Cairo radio announcing that Maghrib prayer is due in Cairo city while I was sitting in my home town at 400 miles North East of Cairo. What is interesting is that at exactly the same time a next door mosque, in my home town, was also calling for the Maghrib prayer. This makes me wonder : how could it be that sunset is simultaneous at two locations separated by a distance of 400 miles from each other and at different Latitudes & Longitudes. As a reminder Maghrib prayer time occurs everywhere at sunset. <br />In what follows we explore this issue and try to prove or disprove the simultaneity of sunset at two different locations. In so doing we are led to some interesting conclusions and as a bonus we got ourselves an animation of the Terminator circle on the surface of the globe. <br />Aside from its modest value and its originality ( I am not aware of anything similar to it ) this article is a good exercise in Maple programming. <br />May this article be a stimulus for some readers to get interested in Astronomy which is a science as ancient as the early human civilizations. <br /><br /></p>100509Tue, 28 Dec 2010 05:00:00 ZDr. Ahmed BaroudyDr. Ahmed BaroudyClassroom Tips and Techniques: Maple Meets Marden's Theorem
http://www.maplesoft.com/applications/view.aspx?SID=99069&ref=Feed
<p>Dan Kalman ascribes to Marden, and describes it as the most amazing theorem: If three noncollinear points in the complex plane forming the vertices of a triangle are interpolated by a (monic) cubic polynomial <em>p(z)</em>, the zeros of <em>p'(z)</em> are the foci of a unique ellipse (the Steiner in-ellipse) that is tangent to the sides of the triangle at the midpoints of the sides. The charm of this easy-to-state theorem that joins algebra, geometry, and complex analysis is explored with the tools of Maple.<br /></p><img src="/view.aspx?si=99069/thumb.jpg" alt="Classroom Tips and Techniques: Maple Meets Marden's Theorem" align="left"/><p>Dan Kalman ascribes to Marden, and describes it as the most amazing theorem: If three noncollinear points in the complex plane forming the vertices of a triangle are interpolated by a (monic) cubic polynomial <em>p(z)</em>, the zeros of <em>p'(z)</em> are the foci of a unique ellipse (the Steiner in-ellipse) that is tangent to the sides of the triangle at the midpoints of the sides. The charm of this easy-to-state theorem that joins algebra, geometry, and complex analysis is explored with the tools of Maple.<br /></p>99069Tue, 16 Nov 2010 05:00:00 ZRobert LopezRobert LopezClassroom Tips and Techniques: The One- and Two-Argument Arctangent Functions in Maple
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<p>In general, a "conversion" between the one- and two-argument artangent functions is not mathematically correct. However, we give two examples of calculations that benefit from a formal interchange of these two functions, and show how different programming strategies in Maple can be used to build tools to make these formal interchanges.</p><img src="/view.aspx?si=97762/thumb.jpg" alt="Classroom Tips and Techniques: The One- and Two-Argument Arctangent Functions in Maple" align="left"/><p>In general, a "conversion" between the one- and two-argument artangent functions is not mathematically correct. However, we give two examples of calculations that benefit from a formal interchange of these two functions, and show how different programming strategies in Maple can be used to build tools to make these formal interchanges.</p>97762Wed, 13 Oct 2010 04:00:00 ZRobert LopezRobert Lopezvan Roomen Problem
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<p>It is a well known fact that sines & cosines of some angles can be expressed in square root radicals.<br />This is the case of sines & cosines of the following angles in degrees : <br />30, 45, 60, 72, 30/2 = 15, 72/2 = 36, 36/2 = 18, 30+18 = 48, 48/2 = 12, 12/2 = 6, 6/2 = 3, 3/2 = 1.5, <br />and all multiples of 3 degrees by a power of 2 are expressible in square root radicals only. <br />This is related to the fact that these radicals can be get from the double of the angle formulas which involve square root radicals only.<br />It is a remarkable fact that all angles counted in degrees as powers of 2 → <br /> n<br /> 2 <br /> such as:<br /> 2deg, 4deg, 8deg, 16deg, 32deg, 64deg, ... , etc.<br />have their sines & cosines which can not be expressed as pure square root radicals but as a combination of cubic & square root radicals.<br />The same holds true for all angles counted in degrees as multiples of 5deg by a power of 2 → <br /> n<br /> 5.2 <br /> such as:<br /> 5deg, 10deg, 20deg, 40deg, 80deg, ... , etc.<br />The purpose of this article is double:<br /><br />1st Purpose - To show how to solve a famous 16 century challenging problem, involving many levels of square root radicals, using Maple powerful calculating engine.<br /> <br />2d Purpose - To find a way to express cosine & sine of any angle from 1 degree on in a combination of cubic & square root radicals. We find first cos(1deg) & sin(1deg). <br />Any other angle being a sum or a difference of two easily found sine & cosine or a multiple by 2 of a an angle.<br /><br />This 2d purpose seems to be an exercise in futility since the final formulas are unwieldy and will never be of any practical use. However the way we tackle the problem is very instructive and possibly many readers may get some insight as to how we can deal with some trigonometry problem the easy way.<br /><br /></p><img src="/view.aspx?si=96978/maple_icon.jpg" alt="van Roomen Problem" align="left"/><p>It is a well known fact that sines & cosines of some angles can be expressed in square root radicals.<br />This is the case of sines & cosines of the following angles in degrees : <br />30, 45, 60, 72, 30/2 = 15, 72/2 = 36, 36/2 = 18, 30+18 = 48, 48/2 = 12, 12/2 = 6, 6/2 = 3, 3/2 = 1.5, <br />and all multiples of 3 degrees by a power of 2 are expressible in square root radicals only. <br />This is related to the fact that these radicals can be get from the double of the angle formulas which involve square root radicals only.<br />It is a remarkable fact that all angles counted in degrees as powers of 2 → <br /> n<br /> 2 <br /> such as:<br /> 2deg, 4deg, 8deg, 16deg, 32deg, 64deg, ... , etc.<br />have their sines & cosines which can not be expressed as pure square root radicals but as a combination of cubic & square root radicals.<br />The same holds true for all angles counted in degrees as multiples of 5deg by a power of 2 → <br /> n<br /> 5.2 <br /> such as:<br /> 5deg, 10deg, 20deg, 40deg, 80deg, ... , etc.<br />The purpose of this article is double:<br /><br />1st Purpose - To show how to solve a famous 16 century challenging problem, involving many levels of square root radicals, using Maple powerful calculating engine.<br /> <br />2d Purpose - To find a way to express cosine & sine of any angle from 1 degree on in a combination of cubic & square root radicals. We find first cos(1deg) & sin(1deg). <br />Any other angle being a sum or a difference of two easily found sine & cosine or a multiple by 2 of a an angle.<br /><br />This 2d purpose seems to be an exercise in futility since the final formulas are unwieldy and will never be of any practical use. However the way we tackle the problem is very instructive and possibly many readers may get some insight as to how we can deal with some trigonometry problem the easy way.<br /><br /></p>96978Sat, 18 Sep 2010 04:00:00 ZDr. Ahmed BaroudyDr. Ahmed BaroudyClassroom Tips and Techniques: Real Distinct Roots of a Cubic
http://www.maplesoft.com/applications/view.aspx?SID=95925&ref=Feed
<p>The real distinct roots of the cubic equation z<sup>3</sup> + a z<sup>2</sup> + b z + c = 0 can be expressed compactly in terms of trig functions. However, Maple's solve command does not return this compact form, so we explore how we can interpret and compact Maple's solution of this equation.<br /><br /></p><img src="/view.aspx?si=95925/thumb.jpg" alt="Classroom Tips and Techniques: Real Distinct Roots of a Cubic" align="left"/><p>The real distinct roots of the cubic equation z<sup>3</sup> + a z<sup>2</sup> + b z + c = 0 can be expressed compactly in terms of trig functions. However, Maple's solve command does not return this compact form, so we explore how we can interpret and compact Maple's solution of this equation.<br /><br /></p>95925Tue, 10 Aug 2010 04:00:00 ZRobert LopezRobert LopezClassroom Tips and Techniques: Trigonometric Parametrization of an Ellipse
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<p>Computing a line integral around an ellipse is more easily done if the ellipse is described parametrically. In this month's article, we delineate how to obtain an exact trigonometric parametrization of an ellipse, even one whose axes are rotated with respect to the coordinate axes.</p><img src="/view.aspx?si=19372/thumb.png" alt="Classroom Tips and Techniques: Trigonometric Parametrization of an Ellipse" align="left"/><p>Computing a line integral around an ellipse is more easily done if the ellipse is described parametrically. In this month's article, we delineate how to obtain an exact trigonometric parametrization of an ellipse, even one whose axes are rotated with respect to the coordinate axes.</p>19372Tue, 07 Apr 2009 04:00:00 ZDr. Robert LopezDr. Robert LopezClassroom Tips and Techniques: Branches and Branch Cuts for the Inverse Trig and Hyperbolic Functions
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Properties of the inverse trig and hyperbolic functions in Maple depend on Maple's choices of branch cuts and principal branches. Some of this information is available to the user, and some can be deduced. In this month's article, we show how to determine principal branches and branch cuts for these twelve functions, and then provide a tool for assembling the information in a user-friendly graphical format.<img src="/view.aspx?si=6932/Untitled-1.jpg" alt="Classroom Tips and Techniques: Branches and Branch Cuts for the Inverse Trig and Hyperbolic Functions" align="left"/>Properties of the inverse trig and hyperbolic functions in Maple depend on Maple's choices of branch cuts and principal branches. Some of this information is available to the user, and some can be deduced. In this month's article, we show how to determine principal branches and branch cuts for these twelve functions, and then provide a tool for assembling the information in a user-friendly graphical format.6932Thu, 27 Nov 2008 00:00:00 ZDr. Robert LopezDr. Robert LopezGraphing interface for A sin(Bx + C) + D
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Provides the student with a command-free environment to experiment with the graph of the sine function in all its glory. Includes sliders for A, B, C, D and radio buttons for selecting radians or degrees. The embedded plot component automatically labels the x-axis in multiples of either Pi/2 or 90 degrees.<img src="/view.aspx?si=6575/1.jpg" alt="Graphing interface for A sin(Bx + C) + D" align="left"/>Provides the student with a command-free environment to experiment with the graph of the sine function in all its glory. Includes sliders for A, B, C, D and radio buttons for selecting radians or degrees. The embedded plot component automatically labels the x-axis in multiples of either Pi/2 or 90 degrees.6575Tue, 26 Aug 2008 00:00:00 ZJason SchattmanJason SchattmanClassroom Tips and Techniques: A Vexing Trig Conversion
http://www.maplesoft.com/applications/view.aspx?SID=6401&ref=Feed
This month's article examines partial solutions that appeared in two issues of MapleTech, one from 1996 and one from 1997. It then provides a complete solution to the transformation coded by one of the Maple developers.<img src="/view.aspx?si=6401/1.jpg" alt="Classroom Tips and Techniques: A Vexing Trig Conversion" align="left"/>This month's article examines partial solutions that appeared in two issues of MapleTech, one from 1996 and one from 1997. It then provides a complete solution to the transformation coded by one of the Maple developers.6401Wed, 02 Jul 2008 00:00:00 ZDr. Robert LopezDr. Robert LopezPlotting of Polar Points
http://www.maplesoft.com/applications/view.aspx?SID=6303&ref=Feed
Given a polar point with its radial and angular component in degrees, the system demonstrates, narrates, and animates the plotting of polar points. It also includes other petinents topics related to polar points.
System is intended for high school or junior college students taking a course in either Trigonometry or Precalculus.<img src="/view.aspx?si=6303/Untitled-1.gif" alt="Plotting of Polar Points" align="left"/>Given a polar point with its radial and angular component in degrees, the system demonstrates, narrates, and animates the plotting of polar points. It also includes other petinents topics related to polar points.
System is intended for high school or junior college students taking a course in either Trigonometry or Precalculus.6303Wed, 21 May 2008 00:00:00 ZProf. P. VelezProf. P. VelezClassroom Tips and Techniques: Stepwise Solution of a Trig Equation
http://www.maplesoft.com/applications/view.aspx?SID=5680&ref=Feed
Stemming from a user's query, we explore how to generate a stepwise solution of a trig equation. Originally, the user wanted to know if the complete solution could be developed in the "Equation Manipulator" but we show that only part of the solution can be obtained this way. A complete solution requires use of a number of Maple's syntax-free features, including palettes, the Context Menu, and the "Equation Manipulator."<img src="/view.aspx?si=5680/Stepwise_Solution_of_a_Trig.gif" alt="Classroom Tips and Techniques: Stepwise Solution of a Trig Equation" align="left"/>Stemming from a user's query, we explore how to generate a stepwise solution of a trig equation. Originally, the user wanted to know if the complete solution could be developed in the "Equation Manipulator" but we show that only part of the solution can be obtained this way. A complete solution requires use of a number of Maple's syntax-free features, including palettes, the Context Menu, and the "Equation Manipulator."5680Thu, 28 Feb 2008 00:00:00 ZDr. Robert LopezDr. Robert LopezAsinchronous Sines Functions
http://www.maplesoft.com/applications/view.aspx?SID=5647&ref=Feed
The properties of the Asinchronous Sines Matrix are described, in
particular: the identity of the determinant for different positions from the sequence.
Note: This is in English and Spanish.<img src="/view.aspx?si=5647//applications/images/app_image_blank_lg.jpg" alt="Asinchronous Sines Functions" align="left"/>The properties of the Asinchronous Sines Matrix are described, in
particular: the identity of the determinant for different positions from the sequence.
Note: This is in English and Spanish.5647Thu, 07 Feb 2008 00:00:00 ZProf. Dante WojtiukProf. Dante Wojtiuk"Just Move It Over There, Dear!"
http://www.maplesoft.com/applications/view.aspx?SID=5158&ref=Feed
My mother once asked me if I could please move her living room sofa into the guest bedroom down the hall and around the corner. Before I broke my back dragging this battleship down the hallway only to discover that it wouldn't make the turn, I decided to take some measurements and work out the math first.<img src="/view.aspx?si=5158/thumb.jpg" alt=""Just Move It Over There, Dear!"" align="left"/>My mother once asked me if I could please move her living room sofa into the guest bedroom down the hall and around the corner. Before I broke my back dragging this battleship down the hallway only to discover that it wouldn't make the turn, I decided to take some measurements and work out the math first.5158Wed, 29 Aug 2007 00:00:00 ZDr. Jason SchattmanDr. Jason SchattmanClassroom Tips and Techniques: Task Templates in Maple
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Maple comes with more than 200 built-in Task Templates. The process of creating a new Task Template, and adding it to the Table of Contents of all such tasks is relatively straightforward. In this article, we explain how to create a new Task Template and add it to the list of built-in tasks.<img src="/view.aspx?si=1763/tasktemplates.gif" alt="Classroom Tips and Techniques: Task Templates in Maple" align="left"/>Maple comes with more than 200 built-in Task Templates. The process of creating a new Task Template, and adding it to the Table of Contents of all such tasks is relatively straightforward. In this article, we explain how to create a new Task Template and add it to the list of built-in tasks.1763Thu, 20 Jul 2006 00:00:00 ZDr. Robert LopezDr. Robert Lopez3-D Tennis Ball
http://www.maplesoft.com/applications/view.aspx?SID=4434&ref=Feed
Ever wonder how tennis ball manufacturers shape the white rubber strip that holds the ball together? This worksheet gives a plausible hypothesis. Starting from the observation that any parametric curve of the form (cos(a(t))sin(b(t)), sin(a(t))sin(b(t)), cos(b(t))) lies on the unit sphere, the goal is to pick a(t) and b(t) so that the curve looks tennisballesque.<img src="/view.aspx?si=4434/tennis.gif" alt="3-D Tennis Ball" align="left"/>Ever wonder how tennis ball manufacturers shape the white rubber strip that holds the ball together? This worksheet gives a plausible hypothesis. Starting from the observation that any parametric curve of the form (cos(a(t))sin(b(t)), sin(a(t))sin(b(t)), cos(b(t))) lies on the unit sphere, the goal is to pick a(t) and b(t) so that the curve looks tennisballesque.4434Mon, 03 Nov 2003 17:17:47 ZAndreas GammelAndreas GammelTrigonometry: Complete Set of Lessons
http://www.maplesoft.com/applications/view.aspx?SID=1411&ref=Feed
This is a set of 11 Maple lessons for high school Trigonometry, developed by Gregory Moore of Orange Coast College. The lessons can also supplement a course in Precalculus or Calculus.
They are designed so you can present each topic as you would during a normal lecture but using Maple as the main presentation tool. The students do not have to understand Maple syntax to benefit from the lessons.
With these Maple lessons, you can carry a topic far beyond what is possible on the blackboard. You can generate a new example or diagram instantly just by changing a few values in the worksheet.
You can show more interesting examples than you could on the board, where the examples always had to be planned to work out "nice". Maple computes the dirty work, so the class can focus on the thinking steps.<img src="/view.aspx?si=1411/trig_logo.gif" alt="Trigonometry: Complete Set of Lessons" align="left"/>This is a set of 11 Maple lessons for high school Trigonometry, developed by Gregory Moore of Orange Coast College. The lessons can also supplement a course in Precalculus or Calculus.
They are designed so you can present each topic as you would during a normal lecture but using Maple as the main presentation tool. The students do not have to understand Maple syntax to benefit from the lessons.
With these Maple lessons, you can carry a topic far beyond what is possible on the blackboard. You can generate a new example or diagram instantly just by changing a few values in the worksheet.
You can show more interesting examples than you could on the board, where the examples always had to be planned to work out "nice". Maple computes the dirty work, so the class can focus on the thinking steps.1411Wed, 01 Oct 2003 00:00:00 ZGregory MooreGregory MoorePost-Secondary Mathematics Education Pack: Complete Set of Lessons
http://www.maplesoft.com/applications/view.aspx?SID=4739&ref=Feed
This package of Maple classroom modules by Gregory A. Moore of Cerritos College is designed to enliven the teaching of mathematics curricula at the high school, community college and beginning university levels. Each of the 49 worksheets, categorized in 13 self-contained modules, supplements a particular lecture topic. The modules cover the full spectrum of topics required for a ground level competence in mathematics.
Supplementing your lectures with these interactive worksheets will open portals to mathematical learning and insight that are simply not possible with chalk alone. They empower the student to experience the beauty of mathematics with less of the drudgery and fear that accompany pure paper and pencil approaches. All concepts are illustrated both algebraically and with interactive color graphics and animations. Each worksheet is ready to use but can also be easily customized by the instructor.
Mr. Moore's essay "Integrating Maple into the Math Curriculum - A Sensible Guide for Educators" (Linked below as Worksheet Output), will guide you through the incorporation of these Maple modules into your classroom instruction.<img src="/view.aspx?si=4739/post_math.gif" alt="Post-Secondary Mathematics Education Pack: Complete Set of Lessons" align="left"/>This package of Maple classroom modules by Gregory A. Moore of Cerritos College is designed to enliven the teaching of mathematics curricula at the high school, community college and beginning university levels. Each of the 49 worksheets, categorized in 13 self-contained modules, supplements a particular lecture topic. The modules cover the full spectrum of topics required for a ground level competence in mathematics.
Supplementing your lectures with these interactive worksheets will open portals to mathematical learning and insight that are simply not possible with chalk alone. They empower the student to experience the beauty of mathematics with less of the drudgery and fear that accompany pure paper and pencil approaches. All concepts are illustrated both algebraically and with interactive color graphics and animations. Each worksheet is ready to use but can also be easily customized by the instructor.
Mr. Moore's essay "Integrating Maple into the Math Curriculum - A Sensible Guide for Educators" (Linked below as Worksheet Output), will guide you through the incorporation of these Maple modules into your classroom instruction.4739Wed, 01 Oct 2003 00:00:00 ZGregory MooreGregory Moore