Create a Powerful Trigonometric Design in Excel

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Here is a Microsoft Excel chart/graphic made for a soul mate by using two birth dates and a lucky number. Make it and have the ability to modify it with your own birthdays and special numbers to make unique designs for special occasions of your own. "What is Trigonometry?" is explained in the Tips Section, as you may find of interest.

Steps

The Tutorial

  1. Create a new Excel workbook with 3 newly named worksheets: Data01, Saves and Chart (unless you are working with Chart Wizard). Below is the image to be created.
  2. Set Preferences. Open Preferences in the Excel menu and follow the directions below for each tab/icon.
    • In General, set R1C1 to Off and select Show the 10 Most Recent Documents .
    • In Edit, set all the first options to checked except Automatically Convert Date System. Set Display number of decimal places to blank(as integers are preferred). Preserve the display of dates and set 30 for 21st century cutoff.
    • In View, click on show Formula Bar and Status Bar and hover for comments of all Objects . Check Show gridlines and set all boxes below that to auto or checked.
    • In Chart, allow show chart names and set data markers on hover and leave the rest unchecked for now.
    • In Calculation, Make sure Automatically is checked and calculate before save is checked. Set max change to .001 without commas as goal-seeking is not done a lot for this workbook. Check save external link values and use 1904 system
    • In Error checking, check all the options.
    • In Save, select save preview picture with new files and Save Autorecover after 5 minutes
    • In Ribbon, keep all of them checked except Hide group titles and Developer .
  3. It helps by placing the cursor at cell A16 and doing Freeze Panes. Place the cursor between the A of column A and the 1 of Row 1 in the top left corner and select the entire worksheet; Format Cells Number Number Decimal Places 4, Font Size 9 or 10.
  4. Enter the Defined Name Variables
    • Into cell A1 input the number 210. 210=109+38+63. 109=round(1954/9/2,0) which is birthday #1, yyyy/m/dd. 38=round(1958/4/13,0) which was birthday #2, being April 13, 1958 described as a double quotient, and 63 is the Lucky Number. It just kept coming up during good events and meaningful moments. Later, substitute your own birth dates of you and your soul mate, or perhaps your parents, or friends, or whomever, and your own Lucky Number or a trial number that makes the design "come out well." In a later step, a constant of .5 is entered and 210/.5 = 420, over 360 rows varying 210 to -210 = exactly 7/6 (420/360 that is). π/6 is 30 degrees since p=180 degrees so 7/6 π = 210 degrees, and 210 is the overall variable number being decremented by 360 degrees versus a cosine and sine function. This sort of even relationship to π between your value in A1 and the constant is wanted in order to get good smooth spherical curves.
    • In cell B1. input the number 360 and Insert Name Define Name it as variable AdjRows. There will actually be 361 rows of calculation, but the formulation depends upon there being 360, as in degrees of a circle. AdjRows is short for Adjusted Rows, the number of rows of input to the final graph form, adjusted by 1 closure row.
    • In cell C1, input the formula (without the quote marks) "=1+((1-SQRT(5))/2-1)", which will result in the value of .618033988749895 being displayed when the cell number is formatted for 14 decimal places. This is the Golden Mean (or Golden Ratio or Proportion) Long Leg, the GMLL. 1 minus the long leg equals the Short Leg and both have been known since Euclid's day. Insert Name Define Name this cell C1 as GMLL. See the Tips section for more information.
    • In cells C7 and D7, type Fact2 and Fact3 respectively. Select area C7:D8 and Insert Name Create Names to create the two variable names Fact2 AND Fact3. and their variables in top row for underneath cells C8 and D8. These variables may also be changed later to arrive at new designs.
    • Input the formula "=Round(1958/4/13,0)" into cell C8, or Fact2, and input "=Fact2" into cell D8 or Fact3. Fact is short for factor. These two variables are factors in the main trigonometric formulas to come. Here, they are both set to the later of the two birth dates.
  5. Enter the following Column Heading Titles into cells A9 to D9: A9: Time, B9: Curves, C9: X, D9: Y. Align center all of these.
  6. Enter the Column Formulas
    • Input into cell A10 "=A1"
    • Edit go to cells A11:A370 and input "=Round(A10-($A$1/AdjRows)*2,14)" into cell A11 and then do Edit Fill Down. This will decrement 210 to -210, a total change of 420 over 360 cells, or 7/6 "time period units" compared to a sphere, in lengths, but also in terms of a particle's distance to travel over time, given the volume is known. See the Tips section for more information.
    • Input .5 into cell B10. Edit Go To cells B11:B370 and enter "=B10" into cell B11 and Edit Fill Down. This will put the constant value of .5 in the column. Set the format of the color of cell B10 to canary yellow so it's recognizable as a variable constant one can change later.
    • Input "=((Sin((A10)/(B10*2)*Fact2*GMLL)*Cos(A10)*Fact2*GMLL)*(Cos((A10)/(B10))*Fact2*GMLL))+Sin(ROW()-10)" into cell C10, select C10:C370 and Edit Fill Down. These are the x values of the graph. They are based upon the formula for a spherical helix in 3D per "CRC Standard Curves" by David von Seggern, modified so that dimension z was modified into dimensions x and y, and the whole was spun about a larger circle. See the Tips Section on Other Websites for more info.
    • Input "=((Sin((A10)/(B10*2)*Fact3*GMLL)*Sin(A10)*Fact3*GMLL)*(Cos((A10)/(B10))*Fact3*GMLL))+Cos(ROW()-10)" into cell D10, select cells D10:D370 and Edit Fill Down. These are the y values for the chart and likewise contain the z values of a 3-dimensional chart.

Explanatory Charts, Diagrams, Photos

  1. Create the Chart
    • Select cells C10:D370 to plot as the chart by selecting the Charts button next, then selecting Chart Option Scatter Smoothed Line.
    • Command c copy the chart and use the plus symbol at bottom of workbook to create a new worksheet. Command v paste it into the new worksheet and drag it 1" down and to right on worksheet. Then select lower right corner and expand the chart a fair amount until the line detail shows clearly.
    • Select Chart Layout Axis. Set Horizontal and Vertical Axes to No Axis.
    • Grab lower right corner of chart and re-size it until it is an approximate square.
    • Double-click on the white plot area and select Gradient, Style Radial, Direction Centered, click on left color tab and select color canary yellow, then right tab and select color fire engine red; tap OK. Adjust until you have bright yellow small center and bright red corners.
    • Double click on the chart's line plot series and set line weight to 1 point. Set color to canary yellow.
  2. Given that your chart resembles the one at the top of this article, you're about done! It helps to save your work. On the Data sheet, select cell range A1:D16 and copy it and activate the worksheet Saves and paste the selected range to the left, then again, a few rows below its bottom, and atop that, do Paste Special Values. You now have saved both the formulas and values that created that particular chart. Activate the Chart and, holding down the Shift key, do Copy Picture. Release the shift key. Activate the Saves worksheet, hold down the shift key again and do Paste Picture. Now you have fulfilled a scientific obligation to keep track of your work. Do this to track changes you make and wish to save.
  3. Save the workbook into an aptly named folder, like "Microsoft Excel Imagery".

Helpful Guidance

  1. Make use of helper article and categories:
    • See the article How to Create a Spirallic Spin Particle Path or Necklace Form or Spherical Border for a list of articles related to Excel, Geometric and/or Trigonometric Art, Charting/Diagramming and Algebraic Formulation.
    • For more art charts and graphs, you might also want to click on Microsoft Excel Imagery, Mathematics, Spreadsheets or Graphics to view many Excel worksheets and charts where Trigonometry, Geometry and Calculus have been turned into Art, or simply click on the category as appears in the upper right white portion of this page, or at the bottom left of the page.

Tips

  • Operators are very important. If the chart looks wrong, make sure that all addition and multiplication symbols are correct, as well as subtraction and division, please.
  • Please leave GMLL in CAPS, else it may not be recognized as the correct variable name. The functions, such as SIN and COS, may be entered in caps, but the variables should go into the formulas just as I've given then to you, or rather, just as you input them.
  • This number, the Golden Mean Long Leg, or GMLL, is used for its quadratic qualities of repeating when squared, proportionately. This lends to the curves a certain precision that is otherwise not possible generally. Even so, some imprecision creeps in and the final numbers are slightly off from the beginning ones. This is fixable perhaps with goal-seeking but it is not needed to be that elaborate for the purposes of image design rather than scientific tokomak design precision here. Insert Name Define Name this cell C1 as GMLL.
  • The volume of a sphere is 4/3 π r^3 and the surface of a sphere is 4πr^2 (or 4 circular areas of πr^2). What we're describing is 7/6 of that. Because of the Theory of Neutral Operators, it is true that 7+ 7/6 = 7 * 7/6 = 49/6 = 8 and 1/6. The theory states that there is a point where the operations of addition and multiplication are held neutral to each other for almost any two numbers a and b, once a or b is known, The relationship is such that for a+b = a*b, b = a/(a-1), so that for a large a, say 10,000, b = almost 1 at 10,000/9,999. It is therefore an asymptotic function and it is used here in the "tokomak design" to converge many rays of energy upon a single source to be fused.
  • "What is Trigonometry?" by Fergus Ray Murray
    • 'Trigonometry is the branch of mathematics that deals with triangles, circles, oscillations and waves; it is absolutely crucial to much of geometry and physics. You will often hear it described as if it was all about triangles, but it is a lot more interesting than that. For one thing, it works with all angles, not just triangles. For another, it describes the behaviour of waves and resonance, which are at the root of how matter works at the most fundamental level. They are behind how sound and light move, and there are reasons to suspect they are involved in our perception of beauty and other facets of how our minds work - so trigonometry turns out to be fundamental to pretty much everything. Any time you want to figure out anything to do with angles, or turning, or swinging, there's trigonometry involved.
    • The first thing to understand with trigonometry is why the mathematics of right-angled triangles should also be the mathematics of circles. Picture a line which can turn around one of its ends, like the hand of a clock. Obviously, the moving end of the line traces out a circle - it's like drawing with a compass. Now, consider how far this point is to the right or left of the centre point (we call this distance x), and how far above or below (which we'll call y). By attaching horizontal and vertical lines of lengths x and y to the ends of the first line we get a right-angled triangle. So the mathematical relationship between circles and the set of right-angled triangles should be clear: The position (x, y) of a point at an angle of θ around a circle of radius r is related to θ and r in exactly the same way that the lengths of the adjacent (x) and opposite (y) sides of a right-angled triangle are related to the length of the hypotenuse r and the angle θ.
    • Sine and Cosine
    • This relationship is expressed by the two most fundamental equations of trigonometry:
    • x = r × cos θ
    • y = r × sin θ Or, equivalently:
    • cos θ = x/r
    • sin θ = y/r
    • Sin (sine) is the ratio of the vertical side (the side opposite the corner we're looking at) to the hypotenuse. Cos (cosine) is likewise the ratio of the horizontal side (the side adjacent to that corner) to the hypotenuse. Sine and cosine are functions, which is to say that they take one number (an angle in this case, usually expressed in degrees or radians) and spit out another. For certain values of θ, it is easy to figure out what the sine and cosine values are going to be just by thinking about what the angle corresponds to on the circle; the simplest cases are for θ = 0°, which is a line pointing right, giving cos θ = 1 and sine θ = 0; a line pointing straight up (ie. θ = 90°), which gives us cos θ = 0 and sine θ = 1, and so on. At 45° the opposite and adjacent sides are the same length, so from Pythagoras' Theorem (r2=x2 + y2) they must each be (√2)/2. For values in between the sine and cosine vary in a smooth curve, so that a plot of sin x against x is your basic wavy line.
    • Cosine is to sine as horizontal is to vertical, so the graph of cosine is just like the graph of sine shifted by one quarter-turn.
  • Tangent
    • The third basic trigonometric function is called the tangent (tan for short), and it is defined as the ratio of the opposite and adjacent sides - that is:
    • tan θ = y/x = sin θ/cos θ Its graph looks like sweeping curved lines between positive and negative infinity.
    • SOH! CAH! TOA!
    • So, to recap - the three main trig functions express the ratios of the sides of triangles like this:
    • sin θ = opposite/hypotenuse
    • cos θ = adjacent/hypotenuse
    • tan θ = opposite/adjacent
    • Inverse Functions and Reciprocals
    • So far, I have only talked about trigonometry as it concerns right-angled triangles and circles. But trigonometry takes in the study of all kinds of triangles - be they equilateral, isosceles or scalene. Equilateral triangles just have three sides the same length, and three 60° corners. Isosceles triangles have two sides the same length and hence two identical angles, so it's easy to split them down the middle and treat them as two identical right-angled triangles back to back. Scalene triangles, on the other hand, have every side and angle different, so if you ever have to calculate their lengths and angles you're likely to want to use the the Sine Rule and the Cosine Rule (unless they happen to be right-angled scalene triangles, which obviously makes things easier). With three different angles to work with, it's easiest to call them A, B and C, and call the lengths of the sides opposite them a, b c. The Sine Rule can then be written:
    • a/sin A = b/sin B = c/sin C
    • This is useful, for instance, if you know two angles and the length of one side of a triangle, and you need to find the length of another side; or if you know the lengths of two sides and one angle (which is not the angle between those sides), and you need to find one or more other angles. In cases where you have two sides and the angle between them, or you are given all three lengths and asked to calculate angles, you will need to switch to the Cosine Rule, which can be written in two main ways:
    • a^2 = b^2 + c^2 - 2 × b × c × cos A or
    • cos A = b^2 + c^2 - a^2/2×b×c
  • The general formula for finding the area of a triangle is
    • area = ½ × base × height which is also equal to
    • area = ½ × a × b × sin C.
    • The choice of which angle is which in all of these equations is of course completely arbitrary, so feel free to swap around a, b and c at will, as long as you also swap A, B and C to make them fit.
    • Slopes and Oscillations
    • Look again at the graphs for sine and cosine; notice that when one is at an extreme of position, the other is at an extreme of slope; this observation is important for several reasons. The slope of the sine curve at any point (which is to say the rate of change of x with respect to θ) is in fact equal to the height of cosine at that point, if the angle is measured in radians - this is one of the reasons mathematicians like radians. Similarly, the slope of the cosine curve at any point is negatively proportional to the sine.
    • This means, if you stop to think about it, that the rate of change of the rate of change at any point (the second differential of a sine or cosine curve, to use the mathematical term) is always in negative proportion to its height at that point; it is as if it was being pushed towards the origin by a force proportional to its distance from it. In fact, in real life when something is pushed towards a central point in proportion to its distance from that point (as in pendulums, weights on springs, molecules trapped in solids, and musical instruments - we call this 'simple harmonic motion') it will indeed move in a sine curve, which is why trigonometry is the mathematics of oscillations as well as triangles and circles.
    • The force on a body in these cases is equal to -k × x where k is a constant depending on the system in question (the spring constant in the case of spring systems) and x is the distance from the equilibrium point; the position of the body at any moment in time is given by
    • x = A × cos (ω × t )
    • where t is time, ω is the angular frequency of the motion, which is equal to k2, and A is the amplitude of the motion.
    • Waves
    • A wave is an oscillation which moves in space, such as sound waves, earthquake waves and the matter waves and light waves which turn out to make up just about everything in the universe. Sine waves turn up all over the place; more complex waveforms can always be broken down into a series of superimposed sine waves of various frequencies, in a process known as a Fourier transform. Sub-atomic 'particles' are best thought of as wave packets.
    • This extremely general applicability of the idea of sine waves results in trigonometric functions turning up everywhere you look in physics. The most general form of the basic wave equation, appearing everywhere from classical mechanics through electromagnetism to quantum physics, is this:
    • x = A × cos (ω × t + d / λ)
    • where λ is the wavelength (the distance between one peak of the wave and the next) and d is the distance along the wave. A full exposition of the mathematics of waves is beyond the scope of this writeup; I will just mention quickly that a fuller understanding of it requires a grasp of the idea of superposition and interference - what happens when waves meet each other; refraction - what happens when a wave passes from one medium to another; and diffraction - what happens when a wave passes through a hole. Standing waves and resonance are also profoundly important almost everywhere that waves turn up; they account for the sounds made by different objects, the energies of photon emitted by different atoms and molecules, and for a staggeringly wide range of other phenomena.'

Warnings

  • If entering one of the long formulas and it won't take, count left and right parentheses to make sure they're properly matched and in their proper places please.

Related Articles

Sources and Citations

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