Compound Angles

Transcript

0:00Introduction

Hello everyone! First, a small challenge. You know sine thirty, forty five and sixty degrees by heart. So what is sine fifteen degrees, exactly, with no calculator? It is not in our table, right? But fifteen is forty five minus thirty. So we need the sine of a difference of two angles. That is today's topic, compound angles, and it comes up often in JEE questions. So, let's begin.

0:30Sine and cosine of a sum or difference

So, here is the first formula. Capital A and capital B are any two angles, and their sum or difference is called a compound angle. This sign, a plus on top of a minus, is read plus or minus: the top sign for A plus B, the bottom sign for A minus B, okay? Sine of A plus or minus B equals sine A cos B, plus or minus cos A sine B, and the sign stays the same. With me? Now let me ask you this: in cos of A plus B, is the sign in the middle plus, or minus? Many students say plus, but careful, it is minus! Cos of A plus B equals cos A cos B, minus sine A sine B, so this sign is upside down: a minus on a plus, read minus or plus. Cos cos, sine sine, and the sign flips. Right? Now back to our challenge. Use the sine formula with the minus sign, and work out sine fifteen degrees; the small circle on screen means degrees. Done? Okay, let's see. Sine fifteen degrees is sine of forty five degrees minus thirty degrees. That is one over root two times root three over two, minus one over root two times one half; the tick-shaped sign is the square root, and the small dot means times. So we get root three minus one, over two root two. And that solves our challenge. Clear? In the same way, with the sign flipped, cos fifteen degrees is root three plus one, over two root two. Got it?

2:05Tangent and cotangent of a sum or difference

Now let's do tan. It comes from dividing the sine formula by the cosine formula, okay? Tan of A plus or minus B equals tan A plus or minus tan B, over one minus or plus tan A tan B, so the sign in the denominator flips again, just like in cos. This holds only where both sides are defined. Tan needs a non-zero cos, so cos A, cos B, and cos of A plus or minus B must all be non-zero; the crossed equals sign means not equal to. Right? Cot of A plus or minus B is cot A cot B, minus or plus one, over cot B plus or minus cot A, with cot B first at the bottom. It also holds only where both sides are defined. With me? Now your turn. Seventy five degrees is forty five plus thirty, tan forty five degrees is one, and tan thirty degrees is one over root three. So what is tan seventy five degrees? Okay, let's check it. Seventy five degrees is forty five plus thirty, so we get one plus one over root three, over one minus one over root three. Multiply the top and the bottom by root three: root three plus one, over root three minus one. Rationalise, and it becomes two plus root three. Very good if you got it! Clear? In the same way, tan fifteen degrees, which is tan of forty five minus thirty degrees, is two minus root three. Got it?

3:36Two product formulas, and three angles

Now two product formulas that save time. Sine of A plus B, times sine of A minus B, is sine squared A minus sine squared B, which also equals cos squared B minus cos squared A, okay? The small two means the whole sine is squared. For cos, the product is cos squared A minus sine squared B, which also equals cos squared B minus sine squared A, so one cos squared and one sine squared each time. Got it? For three angles, tan of A plus B plus C equals S one minus S three, over one minus S two; the small numbers below the S are labels, called subscripts. Right? S one is the sum of the three tans; capital sigma, the big Greek letter, means add over all the angles. S two adds the products of the tans taken two at a time, and S three is the product of all three.

4:30Example 1: sin(A+B), tan(A+B) with A in Q2 and B in Q3

Clear? Now example one. Sine A is three over five, with A in quadrant two; the Roman numeral two means the second quadrant. And cos B is minus twelve over thirteen, with B in quadrant three. Find sine of A plus B and tan of A plus B, okay? Pause the video and try it first. Done? Okay, let's check. Let's find cos A first, from the three four five triangle. It gives four over five, so cos A is four over five, right? No, no, wait, that is the trap! Angle A is in quadrant two, where only sine and cosec are positive, so cos A is minus four over five. Angle B is in quadrant three, where only tan and cot are positive, so from the five twelve thirteen triangle, sine B is minus five over thirteen. With me? Now the sine formula. Three over five times minus twelve over thirteen gives minus thirty six over sixty five. Minus four over five times minus five over thirteen gives plus twenty over sixty five. Together, that makes minus sixteen over sixty five. Okay? For tan, we also need cos of A plus B, where the sign flips. Cos A cos B is forty eight over sixty five, and subtracting sine A sine B, which is negative, adds fifteen over sixty five. So cos of A plus B is sixty three over sixty five. Right? So tan of A plus B is minus sixteen over sixty five, divided by sixty three over sixty five. The sixty fives cancel, giving minus sixteen over sixty three. Clear?

6:15Example 2: A+B+C when tan A, tan B, tan C are 1, 2, 3

Now a harder one. Angles A, B and C are acute, so each is between zero and ninety degrees. Tan A is one, tan B is two, and tan C is three. What is A plus B plus C? Pause the video and try it first. Done? Let's check, with the three-angle formula. S one is one plus two plus three, which gives six. S three is one times two times three, also six. Got it? Now you find S two, the sum of the products taken two at a time. The three products are two, six and three, so S two is eleven. Okay? So tan of A plus B plus C is six minus six, over one minus eleven, that is zero over minus ten, which is zero. Right? So the sum of the three angles is zero, isn't it? No, careful, all three angles are positive, so their sum cannot be zero! Pi, the Greek letter, is one hundred and eighty degrees, and tan A is one, so A is pi over four. Tan increases on acute angles, and tan B and tan C are bigger than one, so B and C lie between pi over four and pi over two; the curved symbol means belongs to, and the round brackets leave the ends out. Adding, the sum lies between three pi over four and five pi over four; the double arrow means implies. With me? Tan is zero only at multiples of pi, and the only one in this range is pi itself. So the sum of the three angles is pi. The lesson: always check where the angle lies. Clear?

7:56Quick revision: sign stays in sin, flips in cos and tan

So, quick revision. Sine of a sum or difference is sine A cos B, plus or minus cos A sine B, and the sign stays. In cos, and in the denominator of tan, the sign flips, and tan works only where defined. And from a tan value, always check where the angle lies. Now try the practice problems on this page. All the best!