Multiple and Sub-multiple Angles

Transcript

0:00Introduction

Hello everyone! Here is a small puzzle. Fifteen degrees is forty five minus thirty, so sine fifteen degrees is easy. But eighteen degrees is not a sum or a difference of thirty, forty five, sixty and ninety degrees. So, can we find sine eighteen degrees exactly, with no calculator? Yes, we can, and you will do it yourself today. The tools are the double, triple and half angle formulas, and they come up often in JEE, so let's begin.

0:32Double angles: put B equal to A

So, take the formula for sine of A plus B, and put B equal to A, okay? The left becomes sine two A, the double angle, and the right becomes sine A cos A, twice. So sine two A equals two sine A cos A, for every angle. Simple, right? Now you find cos two A, by putting B equal to A in cos of A plus B, where the middle sign flips. Yes, exactly! Cos two A equals cos squared A, minus sine squared A; the small two means squared. Since sine squared plus cos squared is one, this also equals two cos squared A minus one, or one minus two sine squared A, as you like. Three forms; pick the one the question needs. With me? Tan two A equals two tan A, over one minus tan squared A, from the tan formula. It holds only where both sides are defined, so cos A and cos two A must be non-zero; the crossed equals sign means not equal to. Clear? Here is a form that saves a lot of time. Write sine two A and cos two A over one, and replace that one by cos squared A plus sine squared A, okay? Now divide the top and the bottom by cos squared A, so cos A must not be zero, and both come out in terms of tan A alone, with one plus tan squared A at the bottom. Plus at the bottom here, but minus in tan two A, so don't mix them up. Got it?

2:00Half angles, and the sign trap

Now turn the double angle around, putting A over two in place of A in the cos two A formulas. Then one minus cos A equals two sine squared of A over two, the half angle. And one plus cos A equals two cos squared of A over two. When you spot one plus cos or one minus cos, think of these two lines. Okay? Write sine A with the double angle formula, divide by one plus cos A from the line above, and cancel. Tan of A over two equals sine A over one plus cos A, and also one minus cos A over sine A, wherever both sides are defined. Clear? From the first line, cos of A over two is plus or minus the square root of one plus cos A, over two. So which sign do we keep? The sign cos has in the quadrant of A, right? No, no, careful, that is the trap! The sign comes from the quadrant of A over two, not of A, okay? Pi, the Greek letter, is an angle in radians here, and pi radians is one hundred and eighty degrees. Say A lies between pi and two pi; the curved symbol means belongs to, and round brackets leave out the ends. Then A over two lies between pi over two and pi, so cos of A over two is less than zero, even when cos A is positive. The double arrow means implies. Got it?

3:23Triple angles

Now triple angles: put B equal to two A in sine of A plus B, and expand. Sine three A equals three sine A, minus four sine cubed A; the small three means cubed. Okay? So cos three A is three cos A, minus four cos cubed A, just like sine, right? No, it is the other way round, and many students slip right here! Cos three A equals four cos cubed A, minus three cos A, so the cube term comes first. With me? And tan three A equals three tan A minus tan cubed A, over one minus three tan squared A, again only where both sides are defined. Clear?

4:08Three products with sixty degrees

Now a lovely result. Theta, the Greek letter, is any angle, and the small circle means degrees. Sine theta, times sine of sixty degrees minus theta, times sine of sixty degrees plus theta, equals one quarter of sine three theta. Right? The cos version is the same, giving one quarter of cos three theta. Okay? But for tan there is no one quarter: the product is just tan three theta, wherever all the tans are defined. Got it? So what is sine twenty degrees, times sine forty degrees, times sine eighty degrees? Which theta do you choose? Theta equals twenty degrees, so sixty minus theta is forty, and sixty plus theta is eighty. The product is one quarter of sine sixty degrees, which is root three over eight. Clear?

4:59Example 1: sin 2A, cos 2A and tan 2A when tan A is 3/4

Now example one. Angle A is acute, between zero and ninety degrees, and tan A is three over four. Find sine two A, cos two A and tan two A, all three. Pause the video and try it first. Done? Okay, let's check, with the tan forms. On top, two times three over four, which is three over two; the small dot means times. At the bottom, one plus nine over sixteen, which is twenty five over sixteen. Three over two, divided by twenty five over sixteen, gives twenty four over twenty five. Okay? For cos two A, the top is one minus nine over sixteen, which is seven over sixteen, over the same twenty five over sixteen. So cos two A is seven over twenty five. Got it? Now tan two A is just double tan A, which is three over two, right? No, no, wait, tan does not double like that! Tan two A is sine two A over cos two A, and the twenty fives cancel. So tan two A is twenty four over seven, defined since cos two A is not zero. With me?

6:08Example 2: exact value of sin 18 degrees, no calculator

Now, remember our puzzle? Let's finish it. Find sine eighteen degrees, using the identity sine two theta equals cos three theta, at theta equals eighteen degrees. Use the double angle on the left and the triple angle on the right. Pause the video and try it first. Done? Okay, let's check. At eighteen degrees, two theta is thirty six degrees, and ninety degrees minus three theta is also thirty six. So sine two theta is sine of ninety degrees minus three theta, which is cos three theta. Okay? Now expand both sides: two sine theta cos theta on the left, and four cos cubed theta, minus three cos theta, on the right. Got it? Cos eighteen degrees is not zero, so divide both sides by cos theta, and let s stand for sine theta. The left becomes two times s, and the right, four times one minus s squared, minus three. With me? Rearrange to get four times s squared, plus two times s, minus one, equals zero. By the quadratic formula, s equals minus two, plus or minus root twenty, all over eight. Clear? Two roots, but sine eighteen degrees has only one value. So which root do we keep? Yes, exactly, the positive one! Eighteen degrees is in the first quadrant, so sine eighteen degrees is greater than zero, and we drop the minus root. Root twenty is two root five, so cancel a two, and sine eighteen degrees is root five minus one, over four. Puzzle solved, with no calculator! Got it?

7:47Quick revision: double, half and triple angle formulas

So, quick revision. Double angles: sine two A is two sine A cos A, and cos two A has three forms. Half angles: the sign of the root comes from the quadrant of A over two. Triple angles: three sine minus four sine cubed, but four cos cubed minus three cos. Now go and try the practice problems on this page. All the best!