Considering all the attention that the moon is getting with the upcoming eclipse this would be a good time to review the possibility of adding new trigonometric functions. I wrote this years ago, in the early days of this blog, but have added more to it including diagrams.
As you may know, the branch of mathematics known as trigonometry deals with triangles. Specifically, right triangles. That is, a triangle containing one 90 degree angle. The internal angles of a triangle always add up to 180 degrees. It is very useful for measurement of the world and universe around us.
Consider a straight line that we will call X. Now let's draw a line perpendicular to X and call it Y. From the point where X and Y meet, the origin, we can draw another line we will refer to as R, for radius. The line, R, can be drawn at any angle out from the origin from 0 degrees to 90 degrees. If it is drawn at 0 degrees, R will be one and the same as the line, X. If R is drawn at 90 degrees, R will be one and the same as the line, Y. If R is drawn at 45 degrees, it will divide the original angle XY into two equal angles.
We can also think of it as a square or rectangle. One side of the rectangle is the X side. The perpendicular side is the Y side. The line, R, is the diagonal line that could be drawn between opposite angles of the rectangle. You may notice that unless R is drawn at either 0 or 90 degrees, it must always be longer than either X or Y. If lines X and Y are equal then R will be 1.414 times as long as either, which is the square root of two.
The length of R, or the radius, as opposed to X and Y, depend on the relative lengths of X and Y. If these two axes are equal, R will form a 45 degree angle to X or Y in the origin. If X and Y in the rectangle are not equal, R will form an angle other than 45 degrees to X or Y.
In trigonometry, we define X as the horizontal axis and Y as the vertical axis. The angle of R is measured from the X axis and intersects the two axes at the origin. We have what we refer to as "trigonometric functions". These are the sine, cosine and, tangent. Each angle has a value for each of these three functions.
The sine is defined as the ratio Y/R for any given angle. The sine starts at zero at 0 degrees and goes to one at 90 degrees. The cosine is defined as the ratio X/R and does the opposite. It starts at one at 0 degrees and decreases to zero at 90 degrees. The tangent is defined as the ratio Y/X for a given angle. It starts at zero at 0 degrees, reaches one at 45 degrees and goes to infinity at 90 degrees.
There are the lesser-used so-called "Inverse Functions". The cosecant is the inverse of the sine. So, it is defined as the ratio R/Y. The secant is the inverse of the cosine and is defined as R/X. The cotangent is X/Y. You may notice that three of the six possible functions have the prefix co- in front of them. These co- functions are the ones whose values decrease as the angle from 0 degrees to 90 degrees increases. The trigonometric functions are based on a 90 degree angle. This, of course, makes sense because in our universe, the dimensions of space form 90 degree angles.
There are two additions that I wish to make to trigonometry. I have noticed that, as useful as trigonometry is, it could be even more useful. The trigonometry that we use now is merely the 90 degree set of functions. The nature of the space we inhabit brings two more sets of meaningful trigonometric functions into being that we are not using as of yet.
45 DEGREE TRIGONOMETRIC FUNCTION
The 90 degree set of functions will always remain the most useful simply because that is the angle at which the dimensions we inhabit intersect. But I have noticed two more useful trigonometric functions. If we multiply the sine by the cosine of an angle from 0 to 90 degrees, we find that the product starts at zero at 0 degrees, peaks at 0.5 at 45 degrees and, goes back to zero at 90 degrees. It would be more convenient to multiply the product by two to have it peak at one at 45 degrees.
What we thus obtain is a trigonometric function of a different set than the traditional functions. This new function is obtained by multiplying two of the original 90 degree functions but it is based on an angle of 45 degrees rather than 90 degrees. It may be true that the spatial dimensions of our universe are based upon an angle that we have defined as 90 degrees. But the nature of this space also causes a number of everyday situations to fit into a description based on a 45 degree function.
To give a few examples of the usefulness of the 45 degree trigonometric function, consider the following. You are at one corner of a rectangle, say an athletic field. You wish to go to the opposite corner of the rectangle. How much travel distance will you save, expressed as a ratio, by cutting diagonally across the field instead of going around the perimeter?
The answer depends on the ratio of the lengths of the two perpendicular sides of the rectangle. If the opposing sides are equal, the direction to the opposite corner will be 45 degrees and the efficiency of the savings will be at a maximum. We can express this efficiency as 1. If the rectangle has one side vastly longer than the perpendicular side so that it is a long thin strip, the efficiency will be much less than 1.
If one side could be infinitely long and the side perpendicular to it was infinitely short, the efficiency of a diagonal crossing would be 0. The efficiency of the diagonal cut, peaking at 1, can be expressed as a ratio, the short side divided by the long side. It is at a maximum when the two perpendicular sides are equal so that the angle of the diagonal is 45 degrees.
At that point, if we multiply the sine of the 45 degree angle by the cosine of the angle, we come up with 0.5 as a result. For convenience, again, we will obtain the Trigonometric Product by performing this multiplication and then multiplying it by a factor of two, so that it starts at 0 and peaks at 1 in the same way as we are accustomed to with the 90 degree functions. It peaks at .5 because 45 degrees is .5 of 90 degrees.
Let's look at another application of the 45 degree function, the Trigonometric Product. Suppose you have a stack of fence panels and you wish to enclose the maximum possible area with these panels. We could refer to the problem as area enclosed per given length of perimeter. The enclosed area would be at a maximum when the two perpendicular sides were equal and would be expressed as short side/long side.
An example of the Trigonometric Product for 45 degree functions is that of a cannon being fired into the air. It provides yet another example. It's range on level ground or sea would be at a maximum when the cannon was aimed equidistant between the horizontal and the vertical, in other words at 45 degrees. It's impact point on the ground could be brought closer than the maximum range by aiming it either higher or lower than 45 degrees. If the cannon were fired straight upward, at 90 degrees, it would theoretically fall right back down on its starting point and it's range would be zero.
If you have ever studied calculus, you may have noticed by now that this 45 degree function can replace quite a bit of calculus and is much simpler and easier. Calculus uses a graphed curve to find maxima and minima of the curve. We seek to find which point on a curve in calculus is the one in which the curve stops it's climb or descent and begins to move in the other direction.
You have seen how the peaking of the Trigonometric Product in the 45 degree function does the same thing. Just picture the zero to one and back to zero again of the function as a calculus graph. And to find such things as distance traveled, it is much easier to calculate the area of a right triangle of certain given angles than it is to find the area under a graph in calculus.
The answer depends on the ratio of the lengths of the two perpendicular sides of the rectangle. If the opposing sides are equal, the direction to the opposite corner will be 45 degrees and the efficiency of the savings will be at a maximum. We can express this efficiency as 1. If the rectangle has one side vastly longer than the perpendicular side so that it is a long thin strip, the efficiency will be much less than 1.
If one side could be infinitely long and the side perpendicular to it was infinitely short, the efficiency of a diagonal crossing would be 0. The efficiency of the diagonal cut, peaking at 1, can be expressed as a ratio, the short side divided by the long side. It is at a maximum when the two perpendicular sides are equal so that the angle of the diagonal is 45 degrees.
At that point, if we multiply the sine of the 45 degree angle by the cosine of the angle, we come up with 0.5 as a result. For convenience, again, we will obtain the Trigonometric Product by performing this multiplication and then multiplying it by a factor of two, so that it starts at 0 and peaks at 1 in the same way as we are accustomed to with the 90 degree functions. It peaks at .5 because 45 degrees is .5 of 90 degrees.
Let's look at another application of the 45 degree function, the Trigonometric Product. Suppose you have a stack of fence panels and you wish to enclose the maximum possible area with these panels. We could refer to the problem as area enclosed per given length of perimeter. The enclosed area would be at a maximum when the two perpendicular sides were equal and would be expressed as short side/long side.
An example of the Trigonometric Product for 45 degree functions is that of a cannon being fired into the air. It provides yet another example. It's range on level ground or sea would be at a maximum when the cannon was aimed equidistant between the horizontal and the vertical, in other words at 45 degrees. It's impact point on the ground could be brought closer than the maximum range by aiming it either higher or lower than 45 degrees. If the cannon were fired straight upward, at 90 degrees, it would theoretically fall right back down on its starting point and it's range would be zero.
If you have ever studied calculus, you may have noticed by now that this 45 degree function can replace quite a bit of calculus and is much simpler and easier. Calculus uses a graphed curve to find maxima and minima of the curve. We seek to find which point on a curve in calculus is the one in which the curve stops it's climb or descent and begins to move in the other direction.
You have seen how the peaking of the Trigonometric Product in the 45 degree function does the same thing. Just picture the zero to one and back to zero again of the function as a calculus graph. And to find such things as distance traveled, it is much easier to calculate the area of a right triangle of certain given angles than it is to find the area under a graph in calculus.
A simple way to illustrate this is multiplying two numbers that sum to a certain number, such as 10. The product will be greatest when the two numbers are equal, 5 x 5 = 25, in the same way that the Trigonometric Product is greatest when the sine and cosine are equal, at 45 degrees where the X and Y axes are equal.
Now that we have the original 90 degree functions and the new 45 degree function, let's add the 180 degree function. Just as the distance across a rectangle involves the 45 degree function, the distance across a circle involves the 180 degree function.
Put another way, the familiar 90 degree functions involve pre-existing equal dimensions. The new 45 degree function involves pre-existing potentially unequal dimensions. The 180 degree functions involve the relationship between lines and pre-existing circles. In 90 degree functions, the radius, R, draws a circle. In the 180 degree function, the circle already exists as our "field" and we draw a chord across the circle.
We could call the original 90 degree functions, the "primary functions" and the new 45 and 180 degree functions, the "secondary functions". Imagine a circle, such as a circular park. Suppose you were at one point on the circle and wished to go to another point on the circle. How much efficiency would you gain by taking a shortcut directly across the circle to the destination point?
The answer would depend on how far ahead was your destination point and thus how much of the circle you were to cut out. The closer the destination point was to the present point, the less would be the efficiency of cutting straight across. The efficiency would be at a maximum if you were cutting directly across the circle to the point diametrically opposite you.
180 DEGREE TRIGONOMETRIC FUNCTION
Now that we have the original 90 degree functions and the new 45 degree function, let's add the 180 degree function. Just as the distance across a rectangle involves the 45 degree function, the distance across a circle involves the 180 degree function.
Put another way, the familiar 90 degree functions involve pre-existing equal dimensions. The new 45 degree function involves pre-existing potentially unequal dimensions. The 180 degree functions involve the relationship between lines and pre-existing circles. In 90 degree functions, the radius, R, draws a circle. In the 180 degree function, the circle already exists as our "field" and we draw a chord across the circle.
We could call the original 90 degree functions, the "primary functions" and the new 45 and 180 degree functions, the "secondary functions". Imagine a circle, such as a circular park. Suppose you were at one point on the circle and wished to go to another point on the circle. How much efficiency would you gain by taking a shortcut directly across the circle to the destination point?
The answer would depend on how far ahead was your destination point and thus how much of the circle you were to cut out. The closer the destination point was to the present point, the less would be the efficiency of cutting straight across. The efficiency would be at a maximum if you were cutting directly across the circle to the point diametrically opposite you.
The proportional distance saving starting at A and going to B would be greater than going A to C in the following diagram. How much greater would be described by the 180 degree function. We can call this trigonometry because the proportional distance savings can be graphed as an isosceles triangle, starting at zero, reaching a peak midway and then going back to zero.
This could also be described in terms of a right triangle in that going from A to C and then to B, instead of directly from A to B, would be 1.414 times as far, with C being at a 90 degree angle from A or B. 1.414 is the square root of 2 and if the X and Y axes of a right triangle are equal in length then the length of the radius, or diagonal, will be 1.414 times either one. Remember the Pythagorean Theorem for a right triangle, A squared equals B squared plus C squared, with A being the diagonal or radius.
The 180 degree function involves the sine multiplied by the cosine that is the basis of the 45 degree function. If we cross a circle by it's diameter as illustrated by the red line in the following diagram and consider the starting point as zero degrees and the destination as 180 degrees, at any point on the diameter the distance to the outside of the circle, by a line perpendicular to the diameter as shown by the blue line in the following diagram, the distance to the outside of the circle is given as a proportion of the diameter by multiplying the sine and cosine of the angle, which is between the zero and 180 degrees.
So, we could say that the expression of efficiency begins at 0 if our destination point is the point immediately ahead of our present position and goes to 1 if our destination point is on the diametrically opposite side of the circle. In other words, 180 degrees ahead. Thus, we have a new trigonometric function that starts at zero for 0 degrees and goes to 1 for 180 degrees.
Such a straight line from a point on a circle, across the circle to another point on the circle, is known as a chord. In a chord, the angle of the outside circle occupied by the chord is equal to the sum of the instantaneous angles formed in the two places where the chord intersects the circle. Obviously, the instantaneous angles formed by chords cannot be over 90 degrees in a circle because 90 multiplied by two equals 180, which is the number of degrees cut off the circle by the longest possible chord, which is a diameter.
By "instantaneous", I mean the immediate angle at only the contact point of the circle. Suppose you had an infinitely large circle. The intersection of a diameter line or a chord would form a clearly measurable angle with the circle. The difference between a chord and a full diameter of a circle is that the instantaneous angles formed in a diameter line are perpendicular, 90 degrees, and in a chord are less than 90 degrees.
This is the basis of the 180 degree function. The instantaneous angle of the radius, R, and the circle thus formed is always 90 degrees in the 90 degree functions. In our new 180 degree function, it varies from 0 to 90 degrees. The instantaneous angle is equal to the angle between the diameter and the chord, which can range from 0 to 90 degrees. Or half the angle of the outside circle that is within the chord, which can range from 0 to 180 degrees.
The length of the radius, R, always stays the same in the 90 degree functions. However, the length of the chord can vary from 0 to 1 in the 180 degree functions. A length of 1 would, of course, be equivalent to a diameter of the circle. As the efficiency becomes greater, the length of the chord becomes longer until at an efficiency of 1, the chord has it's greatest diameter and has become a diameter of the circle. In terms of distance, the efficiency of cutting diametrically across a circle would be 2/pi or 1.57. Now for the actual formula for the 180 degree trigonometric function. It is the sine of the angle, from 0 to 180 degrees.
As we draw a chord of varying lengths across part of a circle from our starting point, we notice that the increase in the length of the chord is much greater from 0 to 90 degrees than it is from 90 to 180 degrees. If a diameter (180 degrees) is considered as having a value of 1, a chord will have a length of .707 when it covers 90 degrees of the circle. It will increase in length only .293 more as it goes from 90 to 180 degrees.
This is simply because as the length of the chord increases in the second quadrant (90-180 degrees), it is decreasing in the first quadrant from .707 when the chord is 90 degrees to 0.5 when it is 180 degrees. At 180 degrees, of course, the chord of 1 will consist of 0.5 length in each of the two quadrants.
This new function is useful whenever there are interior lines forming a chord or a diameter in a circle. For example, two planets in their orbits. Or a specific point on the earth's surface to a particular point on the moon's surface at a particular instant in time. Of course, the orbits of the planets tend to be ellipses instead of perfect circles.
But this function can be easily modified for two ellipses by defining the aphelion and perihelion (furthest and closest points of approach) in terms of relative distance and expressing the orbits as comprising 360 degrees. The difference in horizontal plane of two orbits can easily be expressed in terms of 90 degree trigonometric functions.
To summarize, the 45 degree function describes a radius from an origin consisting of a perpendicular x and y axes. The function equals the sine multiplied by the cosine of the angle of the radius from the horizontal x axis and then multiplied by 2. This function starts at zero at 0 degrees, peaks at 1 at 45 degrees and goes back to zero at 90 degrees. The 180 degree function describes a straight line between two points on the inside of a circle. It starts at zero for an infinitesimal chord and goes to 1 for a diameter, or 180 degree chord.
The efficiency as well as the length of a given chord from zero to the maximum (diametrical) value are both expressed by the same number from 0 to 1. The function is given by the sine of the angle. We could call these two new functions, the "Compound Functions". The sine, cosine and, tangent are the Primary Functions. The cosecant, secant and, cotangent are the Inverse Functions.
The new 180 degree function could possibly be called the "Planetary Function" because it is ideal for describing the directional relationship between two planets, one rotating inside the orbit of the other. Actually the best way to illustrate this function is the view of the moon from the earth. I believe this is also a better way to give an example of trigonometry than anything concerning the traditional 90 degree functions.
The area of the moon that is lighted, as seen from earth, is a function of the angular distance in the sky between the earth and the sun. This can only be described by my 180 degree function and not by any of the 90 degree functions. The proportion of the moon that appears illuminated to us starts at zero at new moon, goes to complete at full moon and then back to zero at new moon. The proportion of the moon that appears as lit from earth is defined as the ratio of Line A to Line B.
Such a straight line from a point on a circle, across the circle to another point on the circle, is known as a chord. In a chord, the angle of the outside circle occupied by the chord is equal to the sum of the instantaneous angles formed in the two places where the chord intersects the circle. Obviously, the instantaneous angles formed by chords cannot be over 90 degrees in a circle because 90 multiplied by two equals 180, which is the number of degrees cut off the circle by the longest possible chord, which is a diameter.
By "instantaneous", I mean the immediate angle at only the contact point of the circle. Suppose you had an infinitely large circle. The intersection of a diameter line or a chord would form a clearly measurable angle with the circle. The difference between a chord and a full diameter of a circle is that the instantaneous angles formed in a diameter line are perpendicular, 90 degrees, and in a chord are less than 90 degrees.
This is the basis of the 180 degree function. The instantaneous angle of the radius, R, and the circle thus formed is always 90 degrees in the 90 degree functions. In our new 180 degree function, it varies from 0 to 90 degrees. The instantaneous angle is equal to the angle between the diameter and the chord, which can range from 0 to 90 degrees. Or half the angle of the outside circle that is within the chord, which can range from 0 to 180 degrees.
The length of the radius, R, always stays the same in the 90 degree functions. However, the length of the chord can vary from 0 to 1 in the 180 degree functions. A length of 1 would, of course, be equivalent to a diameter of the circle. As the efficiency becomes greater, the length of the chord becomes longer until at an efficiency of 1, the chord has it's greatest diameter and has become a diameter of the circle. In terms of distance, the efficiency of cutting diametrically across a circle would be 2/pi or 1.57. Now for the actual formula for the 180 degree trigonometric function. It is the sine of the angle, from 0 to 180 degrees.
As we draw a chord of varying lengths across part of a circle from our starting point, we notice that the increase in the length of the chord is much greater from 0 to 90 degrees than it is from 90 to 180 degrees. If a diameter (180 degrees) is considered as having a value of 1, a chord will have a length of .707 when it covers 90 degrees of the circle. It will increase in length only .293 more as it goes from 90 to 180 degrees.
This is simply because as the length of the chord increases in the second quadrant (90-180 degrees), it is decreasing in the first quadrant from .707 when the chord is 90 degrees to 0.5 when it is 180 degrees. At 180 degrees, of course, the chord of 1 will consist of 0.5 length in each of the two quadrants.
This new function is useful whenever there are interior lines forming a chord or a diameter in a circle. For example, two planets in their orbits. Or a specific point on the earth's surface to a particular point on the moon's surface at a particular instant in time. Of course, the orbits of the planets tend to be ellipses instead of perfect circles.
But this function can be easily modified for two ellipses by defining the aphelion and perihelion (furthest and closest points of approach) in terms of relative distance and expressing the orbits as comprising 360 degrees. The difference in horizontal plane of two orbits can easily be expressed in terms of 90 degree trigonometric functions.
To summarize, the 45 degree function describes a radius from an origin consisting of a perpendicular x and y axes. The function equals the sine multiplied by the cosine of the angle of the radius from the horizontal x axis and then multiplied by 2. This function starts at zero at 0 degrees, peaks at 1 at 45 degrees and goes back to zero at 90 degrees. The 180 degree function describes a straight line between two points on the inside of a circle. It starts at zero for an infinitesimal chord and goes to 1 for a diameter, or 180 degree chord.
The efficiency as well as the length of a given chord from zero to the maximum (diametrical) value are both expressed by the same number from 0 to 1. The function is given by the sine of the angle. We could call these two new functions, the "Compound Functions". The sine, cosine and, tangent are the Primary Functions. The cosecant, secant and, cotangent are the Inverse Functions.
The new 180 degree function could possibly be called the "Planetary Function" because it is ideal for describing the directional relationship between two planets, one rotating inside the orbit of the other. Actually the best way to illustrate this function is the view of the moon from the earth. I believe this is also a better way to give an example of trigonometry than anything concerning the traditional 90 degree functions.
The area of the moon that is lighted, as seen from earth, is a function of the angular distance in the sky between the earth and the sun. This can only be described by my 180 degree function and not by any of the 90 degree functions. The proportion of the moon that appears illuminated to us starts at zero at new moon, goes to complete at full moon and then back to zero at new moon. The proportion of the moon that appears as lit from earth is defined as the ratio of Line A to Line B.
New moon is, of course, when the moon is in the same place in the sky as the sun and full moon is when the moon is 180 degrees opposite the sun. In the above diagram, N is where the moon is at new moon and F at full moon. Take the proportion of the moon that appears lit by the sun, multiply by 180 degrees and that will give the angular distance in the sky between moon and sun.
I believe at this point that there are no more sets of useful trigonometric functions to be found beyond the original 90 degree functions and the new 45 and 180 degree secondary functions that we see here. A meaningful 270 degree (3/4 of a circle) set of functions could not exist because it could not have an equivalent rise and drop and would be a repetition of existing functions. 360 degree functions would be linear and equivalent to 0 degree functions. We could actually define 0 degree trigonometric functions as those functions in only one dimension.
Thus when you measure a straight line or express a linear distance without using trigonometry, you are actually using what we might call the 0 degree function. Although the very definition of the word "trigonometry" means triangle and this is not possible in one dimension. We could, however, refer to measurement of a one-dimensional line as making use of the 0 degree function and say that trigonometry only applies to two-dimensional space.
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