Choosing Between the Quadratic Formula, Sine Rule, Cosine Rule and Area Formula

Picture yourself on the near bank of a river, working out how wide it is without getting wet. You pace out twenty metres along your own side, and from each end of that line you measure the angle across to a tree on the far bank. Two angles and one length, and the width is already fixed. There is only one triangle those three measurements can make. Turning those three numbers into a width takes one formula. Sitting right next to it in your memory are three others that look just as promising, and the question will never tell you which one it wants.

Before any choosing can happen, all four of those formulae need saying out loud, once each and clearly.

All four are Higher tier material. If you are sitting Foundation, none of them will be asked of you, and the video called Recalling Circle, Pythagoras and Trig Formulae is the better use of your time.

First, the quadratic formula. x equals minus b, plus or minus the square root of b squared minus four a c, all over two a. That one is for an equation written as a x squared plus b x plus c equals zero. Second, the sine rule. a over sine A equals b over sine B equals c over sine C. Small a is the side opposite angle A, and the same pattern holds all the way round the triangle. Third, the cosine rule. a squared equals b squared plus c squared, minus two b c cosine A. Notice the squares in it, because that shape is about to earn its keep. Fourth, the area of a triangle: a half times a times b times sine C, where C is the angle sitting between the two sides you just multiplied.

The first sort is free. An x squared in the problem puts you in quadratic territory, and a triangle with no right angle in it puts you among the other three.

The three triangle formulae need one honest test, and that test is all about pairs.

A pair here means a side and the angle directly opposite it, both of them known. The sine rule is built entirely out of pairs: a side over the sine of its opposite angle, set equal to another pair of exactly that shape. Take away one complete pair and the sine rule has nothing left to set equal to anything. The cosine rule is built differently. It ties all three sides to a single angle, so it never needs a matching pair at all.

So here is your handle for this video. Pair it, or square it. If you can find a complete pair, pair it, and that is the sine rule. If you cannot, square it, and the cosine rule is the one with the squares.

First triangle. You know two of its angles and one of its sides, and you want a different side. Sine rule, or cosine rule?

Take your pick. I'll wait.

The answer is the sine rule, and the pair test shows you why. Two angles known means the third is one hundred and eighty minus those two, so every angle is known. Whatever side you were handed, the angle opposite it is known as well, and that is a complete pair.

Second triangle. You know two sides and the angle between them, and the side you want is the one you have not got. Which rule this time?

Pick one. I'll wait.

The answer is the cosine rule, and the test says so before any arithmetic starts. That known angle sits between the two known sides, so it is opposite the missing one. Neither given side has its own opposite angle, so there is no complete pair anywhere.

Grinding either rule through to an actual number is its own topic, and the video on using the sine and cosine rules does that work.

Two triangles, the same two candidates, and the pair test settled both of them without a single calculation.

Now change the question, and leave the triangle exactly where it is.

Same second triangle: two sides, and the angle between them. This time nobody wants a missing side. They want the area of the triangle. Which of the four fires now?

Have a think. I'll wait. The answer is the area formula: a half times a times b times sine C. Two sides with an angle wedged between them is exactly what it feeds on, and it hands you an area in a single line. Look at what just happened there. The same given information pointed at two different formulae, because the thing being asked for changed. What you are given is only half of the decision. Where a half a b sine C comes from is a proof for another day. Here you only need to know that it exists, and what sets it off.

Over in quadratic territory the choice is not about what you are given. It is about what is quick.

On any quadratic, the first move is a quick check for easy factors, before a method gets chosen at all. With a lone x squared out front, you are hunting for two numbers that multiply to give the constant and add to give the middle number.

There are two to try here. Equation a is x squared minus five x plus six equals zero. Equation b is two x squared plus three x minus seven equals zero. Which one of those factorises?

Pause it there and work it out. I'll wait.

Equation a factorises. Two numbers multiplying to six and adding to minus five: minus two and minus three. So it becomes x minus two, times x minus three, equals zero, and the finishing from there is a few seconds of work.

Equation b does not. There is a two in front of the x squared, so multiply that two by the minus seven, then hunt for a pair multiplying to minus fourteen and adding to three. Only four pairs exist to try, and not one of them adds to three.

So equation b is the quadratic formula's job, and that is not a defeat. The formula solves every quadratic, including the ones that were never going to factorise.

Carrying either method through to the solutions belongs to the videos on factorising quadratics and on solving quadratics with the formula.

Check for factors first, and the formula becomes a choice you make rather than a button you reach for.

Now for what the people marking these papers actually say about all of this. This is one line from an OCR examiner report, written about a triangle question that came in two parts. Question 15(b) was a two-stage sine rule question, in that they had to find angle ABC first and then subtract the two angles from 180 to find angle ACB. Some candidates attempted to use the cosine rule but this had been the target of part (a). This question tested when to use these rules as much as testing the use of these rules. Read that last sentence again. The choosing is the question. The rule those candidates reached for was the right rule for the part before it, and familiar is not the same thing as fitting. The same habit turns up on quadratics. Here is a second line, from another OCR examiner report. Even when a quadratic expression could be factorised, many candidates still tried to use the 'formula' or 'complete the square'. Those answers are not wrong. They are slower, and every extra line of arithmetic is one more place for a slip to creep in. So build one habit into every triangle and every quadratic. Before you write anything down, label what you have been given, then name what is being asked for. The formula usually falls straight out of that label.

Right, here is the whole of this video in four decisions. An x squared in front of you? Check for easy factors first, and reach for the quadratic formula when those factors are not there. A triangle with no right angle, and a missing side or angle? Hunt for a complete pair, a side with its opposite angle. Pair it, or square it. Two sides, the angle between them, and the word area? That is a half a b sine C, every single time. Two sides and the angle between them, missing side: cosine rule. Two angles and a side: sine rule. If those came to you before I finished saying them, this one has landed.

Next in the chapter: Using the Kinematics SUVAT Formulae.

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