The cosine rule

Push three sticks together at their ends, eight, eleven and fourteen centimetres long. They make one triangle and only one, so its three angles are already fixed. No angle is given, so the sine rule has no complete pair to start from. There's no right angle either, so Pythagoras and SOH CAH TOA can't help. How big is the corner facing the fourteen-centimetre stick? One rule gets there in three moves, and its first job is finding a side, not an angle.

This is video seven of eleven in Pythagoras, Trigonometry and Coordinate Geometry. It uses the capital and small letters from The sine rule and the area of a triangle, so start there if those letters are new.

The cosine rule is set on Higher papers, and on Extended papers for courses that split into Core and Extended.

The sine rule needs a complete pair: a side and the angle it faces, both known. When a triangle has no complete pair, the cosine rule takes over. That happens in two ways. Either you know two sides and the angle between them, or you know all three sides and no angle at all. The cosine rule says: a squared equals b squared plus c squared, minus two b c cos A. Small a is the side standing on its own, and capital A is the angle it faces. Small b and small c are the other two sides, and angle A is the corner where they meet. On some papers this rule is printed on a formula sheet, so check whether yours is one. What a question tests is putting the right numbers in, and rearranging. An examiner's report on a Higher paper says this. "The students are reminded that the formula for the cosine rule and the sine rule is given on the formula sheet as some students quoted the cosine rule and or the sine rule incorrectly." Those students wrote a rule down wrongly, when it was printed right in front of them. The fix is to copy it from the sheet, letter for letter, before a single number goes in. Now make angle A ninety degrees. What does the cosine rule turn into? It turns into Pythagoras. Cos ninety is nought, so the two b c cos A part vanishes, and a squared equals b squared plus c squared. So the cosine rule is Pythagoras with a correction. Close angle A below ninety, and the side facing it gets shorter than Pythagoras would give, so the rule takes two b c cos A away. Past ninety, cos A is negative, and taking away a negative adds on. The side facing an obtuse angle comes out longer than Pythagoras would give.

A triangular plot of land has two sides of fifteen metres and twenty-two metres, with an angle of sixty-three degrees between them. The third side is wanted, to one decimal place. The sixty-three degrees sits between the two known sides, so it's angle A. The side it faces, the one you want, is a. That makes b fifteen metres and c twenty-two metres. Does it matter which of the fifteen and the twenty-two you call b? No. In the rule, b and c are only ever squared and added, or multiplied together, and the order doesn't change either of those. Right, put the numbers in. a squared equals fifteen squared plus twenty-two squared, minus two times fifteen times twenty-two times cos sixty-three. Fifteen squared is two hundred and twenty-five. Twenty-two squared is four hundred and eighty-four. Added together, that's seven hundred and nine. Next, two times fifteen times twenty-two is six hundred and sixty. That leaves a squared equals seven hundred and nine, minus six hundred and sixty times cos sixty-three. The six hundred and sixty is multiplied by cos sixty-three first, and only then taken away. Seven hundred and nine minus six hundred and sixty, done first, is the wrong order. Press seven hundred and nine, minus, six hundred and sixty, times, cos, sixty-three, close the bracket, equals. The display reads four hundred and nine point three six six two seven zero two. Careful - that's a squared, not a. Press the square root key, then the answer key, equals. The display reads twenty point two three two eight zero one eight four. To one decimal place, the plot's third side is twenty point two metres.

Now, the three sticks give three sides and no angle. The angle facing the fourteen-centimetre stick is the one wanted. The wanted angle is A, so the side it faces, fourteen centimetres, is a. The eight and the eleven are b and c. You don't need a second formula for angles, though. Start from the rule on the sheet and rearrange it, one move at a time. a squared equals b squared plus c squared, minus two b c cos A. First, add two b c cos A to both sides. That gives a squared plus two b c cos A equals b squared plus c squared. Next, take a squared away from each side. That leaves two b c cos A equals b squared plus c squared, minus a squared. One move is left. What do you do to get cos A on its own? Divide by two b c, the number multiplying cos A. Cos A equals b squared plus c squared, minus a squared, all over two b c. Read it as a picture. The side facing the angle is the one taken away on top, and the two sides that meet at the angle are multiplied on the bottom. A different examiner's report, on a question where an angle was wanted, notes this. "Many students did know to use the cosine rule, but often chose to use a rearranged form to give the angle directly." A form written from memory is easy to get wrong, and nothing on your paper checks it. The fix is the three moves you've just made: add, take away, divide. Now the sticks' numbers. Eight squared is sixty-four, eleven squared is a hundred and twenty-one, and fourteen squared is a hundred and ninety-six. On top, sixty-four plus a hundred and twenty-one is a hundred and eighty-five. Take away a hundred and ninety-six, and that's minus eleven. On the bottom, two times eight times eleven is a hundred and seventy-six. So cos A is minus eleven over a hundred and seventy-six. Press minus eleven, divide, a hundred and seventy-six, equals, and the display reads minus one over sixteen. That cosine is negative. What does it tell you about angle A, before any more keys? It's obtuse, bigger than ninety degrees, because cosine turns negative only past ninety. Now press shift, cos, the answer key, equals. The display reads ninety-three point five eight three three two one seven. To one decimal place, that's ninety-three point six degrees. It's obtuse, as the minus sign said it would be, and it faces the longest stick, as an obtuse angle always does. Write it with its name, never as a bare number: angle A, the corner facing the fourteen-centimetre side, is ninety-three point six degrees. A right number beside the wrong corner is still wrong.

A four-sided field A B C D has a straight path from B to D, splitting it into two triangles. In triangle A B D, side A B is ten metres, side A D is twelve metres, and angle B A D is forty-two degrees. In the other triangle, angle B D C is fifty-seven degrees, and the length of the path B D is wanted. Student A writes B D squared equals ten squared plus twelve squared, minus two times ten times twelve times cos forty-two. Student B writes the same line with cos fifty-seven. Which student put the right angle into the rule, and why? Student A. The rule needs the angle where the two known sides meet, which is the corner at A. Every angle faces its own side - look straight across. Stand in the forty-two degrees at A and you look straight across at B D, the side you want. Stand in the fifty-seven at D and you're looking at B C instead. Follow Student A's line through. Ten squared plus twelve squared is two hundred and forty-four, and two times ten times twelve is two hundred and forty. That leaves B D squared equals two hundred and forty-four, minus two hundred and forty times cos forty-two. Press two hundred and forty-four, minus, two hundred and forty, times, cos, forty-two, close the bracket, equals. The display reads sixty-five point six four five two four one eight nine. Press the square root key, then the answer key, equals, and the display reads eight point one zero two one seven five one three. To one decimal place, the path B D is eight point one metres. Student B's line gives two hundred and forty-four, minus two hundred and forty times cos fifty-seven, about a hundred and thirteen point three. Its square root is about ten point six metres, the wrong length for the path. An examiner's report on a cosine rule question records the slip. "The most common error was to use an angle of fifty-seven degrees rather than forty-two degrees, limiting the award to two marks." Fifty-seven degrees was sitting in the diagram, close by. The fix is to stand in the angle before cos goes in, and check it looks straight across at the side you want.

The sticks, in the end, gave up their angle once the rule was rearranged. Time to test it on triangles you haven't met. Starting from the rule for a side, what is cos A equal to? Cos A equals b squared plus c squared, minus a squared, all over two b c. Three moves from a squared equals b squared plus c squared minus two b c cos A. Now try sides of five and eight metres meeting at sixty degrees. How long is the third? Seven metres exactly. Twenty-five and sixty-four make eighty-nine. Two times five times eight times cos sixty is forty, and eighty-nine take away forty is forty-nine, whose root is seven. One more: sides of three, five and seven centimetres. Is the angle facing the seven obtuse? Yes. Three squared and five squared come to thirty-four, less than seven squared, so the top is negative. Cos is minus fifteen over thirty, minus a half, which makes a hundred and twenty degrees.

If you've got it, give this one a thumbs-up, and it comes off the list of videos you still need. If not, save it for later. On some papers the rule itself is printed for you, so substituting and rearranging are the parts to rehearse.

Next in the chapter: Pythagoras' theorem in three dimensions.

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