The Wonder and Amusement of Triangles - Part Three : the Sine Rule
As in the Bible, and in Government, we are subject to certain rules and laws which govern our behaviour, and certain aspects of mathematics are no exception. Now although certain man made laws may be broken, although a good deal of them should not be, those in the realm especially of Mathematics and Physics are inviolate, and it is not like we can pick and choose whether we agree with a Theorem that has been proven beyond doubt and established as fact for hundreds, if not thousands of years.
There are rules for everything in the world of numbers, and if we ignore them, we end up mistaken. Thus there is no surprise that all three of the main Trigonometric functions are bounded by certain Laws, which allow calculation of angles and lengths, missing or otherwise, and make the task a lot easier.
This is so when we use these functions for other triangles, which do not have right angles in them. Now because any triangle can be formed into a collection of right angled triangles, from which we obtain our Trigonomentric ratios, we can find out lengths and angles within any triangle, based upon certain laws.
First Stop - the Sine Rule
As noted, not every triangle has a right angle in it, but even then, it is still possible to work out all the angles and sides if we know certain things. But in saying that, such things need to be known, since as noted in the previous Hub The Wonder and Amusement of Triangles - Part Two, the Law of Missing Lengths, if we just show a triangle, and absolutely no lengths, especially with respect to what units the lengths are in, what can we know ? Say we even had three lengths, but did not know if the lengths referred to inches, feet or miles, now sure, from these we can find all angles, as will be shown, but the actual size will be a mystery. Sometimes though in Maths we do designate the sides of shapes in units of something, whatever these units are, without knowing how large such unit lengths are meant to be, but even from that we know how comparable each length is to the others. In the last Hub I drew an example triangle in feet, but for this I shall express my lengths in inches, so we get an idea of the size we are dealing with.
Sine Rule Diagram by Chris Lilly on Paintbox™
Here we see a triangle with sides A, B, and C, and angles opposite these given as α ( alpha ), β ( beta ), and θ ( theta ), respectively. Logically, the longest side is opposite the largest angle, and the Sine Rule states that when the sine of an angle is divided by the length of its opposite side, this value is the same for all three angles and sides within the same triangle - including right angled triangles.
This can be shown in the diagram below, and as noted, it holds for the three reciprocal values.
The first three are used if we want to find at least one of the angles, when we might know the side opposite the angle in question, and one of the other angles, with the side opposite it.
The reciprocals are used to find the lengths of sides in question, where in both cases, we can rearrange equations to get what we want.
The Sine Rule by Chris Lilly on Paintbox™
As an Example, drawn by Chris Lilly on Paintbox™
How it works
Imagine in the diagram above, that angle α = 65°, and that the side opposite it, side A, = 3”, and in addition, angle β = 40°, and angle θ = 75°. Using the Sine Rule, how do we find out the lengths of both of the other of sides ?
Sine Rule in Action, by Chris Lilly on Paintbox™
Because as B can be divided by sine 40° to find 3 ÷ sine 65°, it stands to reason that we can also multiply sine 40° by 3 ÷ sine 65° to find B. Now, 3 ÷ sine 65° is equal to 3.31, and this multiplied by sine 40° is now equal to 2.1277”, which equals B. To find the third length, we then multiply sine 75° by 3 ÷ sine 65°, ( = 3.31 ), to get a value for C of 3.197”. This should be reasonably straightforward, and the key is to practice, and visualise what You are doing when rearranging equations, and why You are doing it in that way.
This Time it will be Different
Imagine this time that we have a triangle with sides of two, three, and four inches, respectively. Now with the Sine Rule alone we cannot solve this triangle without knowing at least one of the angles, even though we can rearrange these sines and lengths into all sorts of relationships, which don’t help us if we cannot isolate a single sine with a given value.
Sine Rule Example by Chris Lilly on Paintbox™
Theoretically . . .
Simply for example, let us call angle β, which is opposite the side that is 3” long, equal to 46° 34’ 03”, or 46.5675°, which indeed it is, but I found this using the Cosine Rule we will look at later, just to give us this example to work with. Now it is not necessarily the case that any old value will do, for as we shall see with the Cosine Rule, if we have three sides all of known values, then their angles have to take certain values in order for everything to be consistent, since the angles are found only be taking all three side lengths into consideration, thus, they cannot work any other way. Given that the angle stated is actually the right value, let us see what the other two should be. So here we go : Sine 46.5675° ÷ 3 = sine α ÷ 2 = sine θ ÷ 4
By the same logic we used before, if sine 46.5675° ÷ 3 = sine α ÷ 2, this means that we can multiply sine 46.5675° ÷ 3 by 2, in order to find the sine of α.
It goes like this : 2 sine 46.5675° ÷ 3 = sine α, which is .484.
What we do now is use the arcus sine function on our scientific calculator. [ Again, as I have said before, this has been wrongly labelled on calculators as sin-1, since that would indicate a reciprocal, ( 1 ÷ sine ), which the arcus function is not. Rather, an arcus sine function is basically the angle that has the sine that we’ve been given, or have worked out ].
Now, the arcus sine function of .484 = 28.955°, so the angle α equals this 28.955°.
The Next Step
So this is the second angle. Now we can work out the third in either of two ways : we can use the sine rule yet again to find the third one, or we could add the previous two together, then subtract this number from 180° to give it to us, knowing that the internal angles of a triangle do add up to 180°.
Sine Rule : sine 46.5675 ÷ 3 = sine θ ÷ 4, therefore 4 sine 46.5675 ÷ 3 = sine θ
What this means, is that this time, we can multiply sine 46.5675 ÷ 3 ( = .242 ) by four to find the sine of this last angle, then having done so, work out its arcus sine function.
Thus : sine 46.5675 ÷ 3 ( = .242 ) × 4 = .9682. arcus sine function of .9682 = 75.52°.
Aah, but the trouble is, this doesn’t seem right. Roughly adding all three angles so far is nowhere near 180°. In fact, if the third one is added as it is , we get a total of about 150°, which is thirty degrees short.
This is disastrous ! What outrage is this ? Actually, it is all because the arcus sine function on a calculator only works for angles up to 90°, and from here we have to apply a little Circular Trigonometry to work this. We shall have a look at this later, but for now it suffices to say that the sine function can be plotted on the Cartesian Axes Plane as a repeating wave. This wave has positive y values when the sine is between 0° and 180°, and is mirrored about x = 90°. What we do, though, is we subtract from 180° the value of the angle we got from the arcus sine function, and this will give us our real value of approx. 104. 4775°, which it does appear to be so on the diagram above.
A Faux Pas
So what indeed if we wanted to put just any old angle in ? Say again we had the triangle in the diagram from before, but said that one of the angles, the one opposite the three inch long side, was say exactly forty degrees. Then based on the Sine Rule,
sine 40° ÷ 3 = sine α ÷ 2 = sine θ ÷ 4 → sine α = 2 sine 40° ÷ 3 = .428525
Now, arcsin ( .428525 ) = 25.374° ( 3 d.p. )
But then, sine 40° ÷ 3 = sine θ ÷ 4 → sine θ = 4 sine 40° ÷ 3 = .857,
and arcsin ( .857 ) = 58.987° ( 3 d.p. ), but the sum of these three angles only gives us about 124°, and even taking into consideration the symmetry of the Sine Function, this is not right. If we think about how triangles are formed, imagine we did have a forty degree angle opposite a side equalling three inches, and one of the other sides was equal to two inches, these facts then limit what the last side must be, but it should not be equal to four inches, as it would be with the other angle given in the previous example. The next aspect of this we shall look at, which adds to our ability to solve triangles, and can be used alone or in conjunction with other laws, is what is known as The Cosine Rule, and this we can use to shed light on this problem here, as well as other ones.
The Burden of Proof
So, this is all very well, but where's the Proof ? Now I know there are other proofs in certain Maths books of this Rule, but I thought I would look at doing my own. This I managed from scratch, using already known Mathematical principles, and some logic.
Diagram to help show Proof of the Sine Rule, by Chris Lilly on Paintbox™
From this, Algebraic proof of the Sine Rule, also by Chris Lilly on Paintbox™
As a Consequence : (Illustration by Chris Lilly on Paintbox™)
So the Sine Rule has been shown and proven, and interestingly, it has helped in the proof of another theorem, that as well as being ½ base × height, the Area of a Triangle is also expressed by ½ AB Sin θ, which itself can be also seen in a way to relate to our next adventure, the demonstration and proof of the Cosine Rule, since it involves the use of two sides and the angle made by them.
As much as some of this Hub contains certain Mathematical knowledge accessible in the public domain, and not subject to any Copyright, other information has been drawn from textbooks which themselves are Copyright, but only in the sense of how they deliver the information which itself is shared and sometimes ancient Mathematical knowledge. Other information, and the illustration of the illustrious Ana Ivanovic, has also been found on Wikipedia ( Copyright 2013, the Wikimedia Foundation ), which is a good source of information. Part of this is also my own discovery, but may also independently have been found by others.
The Title This Time it will be Different seems to be the Title of works by other authors, namely Susan Santoro, and another which says Today Will Be Different, by Maria Semple. It is also a quote from the newspapers, and now I am not sure where I got my Title from. Some of the illustrations in this particular Hub are my own, and have primarily been done using Microsoft™ Paintbox, edited from illustrations done using Microsoft™ Word. Any quote or part of this material which seems to belong to any other author should be treated as such, and I claim no ownership of anything I did not myself invent or discover, nor of any obvious copyright, trade mark, or registered trade mark.
The Adventure continues in the next Hub, which carries on the study of Triangles, and this one is : The Wonder and Amusement of Triangles - Part Four : the Cosine Rule.
If You are curious, then do not hesitate to take a good look at the other Hubs, The Maths They Never Taught Us - Part One, The Maths They Never Taught Us - Part Two , The Maths They Never Taught Us - Part Three, The Very Next Step - Squares and the Power of Two , And then there were Three - a Study on Cubes, Moving on to Higher Powers - a First look at Exponents, The Power of Many More - more on the Use of Exponents, Mathematics - the Science of Patterns , More on the Patterns of Maths , Mathematics of Cricket , The Shape of Things to Come , Trigonometry to begin with , Pythagorean Theorem and Triplets, Things to do with Shapes, Pyramids - How to find their Height and Volume, How to find the Area of Regular Polygons, The Wonder and Amusement of Triangles - Part One, and The Wonder and Amusement of Triangles - Part Two, the Law of Missing Lengths.
Also, feel free to check out my non Maths Hubs :
There will also be many More to come on a wide variety of Subjects Just take a good look at it, and note how interesting it all is, then see if you can come up with anything else along the same lines. As usual, I would appreciate any comments, feedback and suggestions which would be given due credit, or indeed have a go and publishing Your ideas Yourselves, but firstly, by all means, add Your comments - it's a free Country.