Sum and Product Formulae

Transcript

0:00Introduction

Hello everyone! Let us start with a small puzzle. Cos twenty degrees, plus cos one hundred degrees, plus cos one hundred and forty degrees. Not one of these angles is in our table of standard values, right? So can we find this sum exactly, with no calculator? Believe it or not, the answer is exactly zero. Today we will see why, with the sum and product formulae. They turn a sum of sines or cosines into a product, and a product back into a sum, and they come up often in JEE questions. So, let's begin.

0:35Sum to product: two sines

So, here is the first formula. Capital C and capital D are any two angles. Sine C plus sine D equals two sine of C plus D over two, times cos of C minus D over two. The fraction bar is read as over, so C plus D over two is the half-sum, and C minus D over two is the half-difference, okay? Now let me ask you this: for sine C minus sine D, what do you think changes? The sine and the cos swap places, that is the change! Sine C minus sine D equals two cos of the half-sum, times sine of the half-difference. So for a sum, sine comes first, and for a difference, cos comes first. With me?

1:21Sum to product: two cosines

Now the cosines. Cos C plus cos D equals two cos of the half-sum, times cos of the half-difference. Cos and cos, the easiest one of all. Clear? So what will cos C minus cos D be? Two sine of the half-sum, times sine of C minus D over two, right? Almost, but careful, look at the last angle! Cos C minus cos D equals two sine of the half-sum, times sine of D minus C over two. D comes first here, then C, okay? Why does the order matter so much? Because sine of D minus C over two is minus sine of C minus D over two; changing the sign of an angle changes the sign of its sine. So if you write C minus D there, the sign of your whole answer flips. This is the classic trap in this topic. Got it?

2:16Product to sum: the same formulas, read backwards

Now the other direction: a product into a sum. Two sine A cos B equals sine of A plus B, plus sine of A minus B; the round brackets hold the angle. It is just the first formula read backwards, with C as A plus B, and D as A minus B, right? Next, two cos A sine B equals sine of A plus B, minus sine of A minus B, so this time we subtract. With me? Two cos A cos B equals cos of A plus B, plus cos of A minus B, nice and simple, okay? Now the last one, and it hides a trap. What is two sine A sine B? It is cos of A minus B, minus cos of A plus B, so the difference angle comes first! Many students write cos of A plus B first, and then the sign of the answer comes out wrong. Clear?

3:07Topper's trick: pair the terms

Now let me share a trick that toppers use. When a question adds three or four sines or cosines, do not open them one by one. Pair the terms, so that a common factor appears. Got it? Usually we pair the first with the last, and the second with the second-last, so that the half-sums match. So take sine A, plus sine three A, plus sine five A, and tell me: which two terms should we pair here? Yes, exactly, the first and the last! Sine A plus sine five A equals two sine three A, times cos two A, okay? The half-sum is three A, the same angle as the middle term, so sine three A comes out as a common factor. That's the trick.

3:52Example 1: sin 75 - sin 15, and cos 20 + cos 100 + cos 140

Now example one. First, find sine seventy five degrees minus sine fifteen degrees; the small circle on screen means degrees. Then, our puzzle: cos twenty degrees, plus cos one hundred degrees, plus cos one hundred and forty degrees. Pause the video and try it first. Done? Okay, let's check. Sine minus sine, so cos comes first. The half-sum is ninety over two, that is forty five, and the half-difference is sixty over two, that is thirty. So we get two cos forty five degrees, times sine thirty degrees. Clear? Both are standard values; the tick-shaped sign is the square root, and the small dot means times. Two times one over root two, times one half, gives one over root two. Okay? Now the puzzle, with three cosines. Which two should we pair, so that a nice angle appears? Pair cos one hundred degrees with cos one hundred and forty degrees. The half-sum is two hundred and forty over two, which is one hundred and twenty, a standard angle! The half-difference is minus twenty degrees, and cos of minus twenty degrees is just cos twenty degrees, because the cos of a negative angle equals the cos of the positive angle. With me? Cos one hundred and twenty degrees is minus one half, shown inside the brackets. So two times minus one half, times cos twenty degrees, gives minus cos twenty degrees. Right? Now bring back the first term. Cos twenty degrees, minus cos twenty degrees, is zero! Puzzle solved: exactly zero, with no calculator. See how the pairing made the common factor appear? Got it?

5:40Example 2: sin 12 sin 48 sin 54 using product to sum

Now a harder one, JEE level. Find sine twelve degrees, times sine forty eight degrees, times sine fifty four degrees. Three sines multiplied together, none of them in our table. Pause the video and try it first. Hint: use a product to sum formula. Done? Okay, let's check. Which two sines should we combine first? Sine twelve and sine forty eight, because their difference is thirty six and their sum is sixty, both nice angles! Use two sine A sine B, with the difference first. So sine twelve degrees times sine forty eight degrees equals one half, times cos thirty six degrees minus cos sixty degrees. Clear? Now look at sine fifty four degrees. Fifty four plus thirty six is ninety, so sine fifty four degrees equals cos thirty six degrees. Let us call it small c, to keep the writing short. With me? Cos sixty degrees is one half, so the whole product is one half, times the bracket c minus one half, times c. Opening the bracket, that is one half of c squared minus c over two; the small two up top means squared. Okay? Now, which special value do we need here? What is cos thirty six degrees? It is root five plus one, over four, from the special values of the previous lesson. Squaring it, c squared is three plus root five, over eight. And c over two is one plus root five, over eight. Right? Now subtract. The root fives cancel, and three minus one leaves two over eight, which is one quarter. So the product is one half times one quarter, which gives one over eight. Three sines that are not in any table, and the answer is just one eighth. Beautiful, isn't it?

7:34Quick revision: the two sign traps and the pairing trick

So, quick revision. Sum to product: use the half-sum and the half-difference, and in cos C minus cos D, the last angle is D minus C, in that order. Product to sum: two sine A sine B is cos of A minus B, minus cos of A plus B, with the difference first. And with many terms, pair the first with the last to get a common factor. Now go and try the practice problems on this page. All the best!