Allied Angles, Periodicity and Graphs

Transcript

0:00Introduction

Hello everyone! Let us start with a small puzzle. Mod sine x, the size of sine x without its sign, repeats every pi. So does mod cos x. After how long does their sum repeat? Surely it is pi, right? Hold that thought, the answer may surprise you. Allied angles and periods come up often in JEE, so let's begin.

0:25Allied angles: the two-step rule

So, here is the big idea: any angle n pi over two, plus or minus theta. Here n is an integer, and the sign with a plus over a minus means plus or minus. The rule works for any theta, but to find the sign we treat the Greek letter theta as acute. Okay? Step one decides the function. If n is even, the double arrow points to the answer: the same function. Sine stays sine, and cos stays cos. Clear? If n is odd, the function switches to its partner: sine and cos, tan and cot, sec and cosec. That is what the arrows with two heads mean. Right? Step two decides the sign. Find the quadrant of the whole angle, with theta acute, and attach the sign of the original function there, the one you started with. All Silver Tea Cups, from the last lesson, gives the signs. With me?

1:18Two quick tests, and a trap

Let's test the rule on cos of pi plus theta. Pi is two pi over two, so n is two, even, and cos stays cos. Now, in which quadrant is pi plus theta, and what sign does cos have there? It is the third quadrant, yes, exactly! There only tan and cot are positive, so cos is less than zero. So, cos of pi plus theta equals minus cos theta. Got it? Now a trickier one: cos of pi over two plus theta. Here n is one, odd, so cos becomes sine. The angle is in the second quadrant, where sine is positive. So the answer is plus sine theta, right? Oh no, careful, that is the trap! The sign comes from the original function, cos, not from the new one. In the second quadrant cos is negative, so cos of pi over two plus theta equals minus sine theta. Clear? For minus theta, n is zero, and the angle is in the fourth quadrant. So sine of minus theta is minus sine theta, and cos of minus theta is just cos theta. Okay?

2:26Periodic functions and their graphs

Now the second big idea: periodic functions. We call f periodic if, for some T greater than zero, f of x plus T equals f of x for every x, meaning every x in the domain of f. The least such T is the period. Right? Look at the graphs. The solid dark blue curve is sine x, and the dashed orange curve is cos x. After how long does the sine curve start repeating itself? It repeats after two pi, yes, exactly! Sine and cos have period two pi, and so do sec and cosec. And the cos curve is the sine curve shifted left by pi over two. With me? Now the green graph, tan x. It repeats every pi, so tan and cot have period pi. The dashed vertical lines sit at odd multiples of pi over two, where tan is not defined.

3:19New periods from old ones

Got it? Now, if f has period T, then f of, a times x, plus b, has period T over mod a. The two vertical bars mean modulus, the size without the sign. Adding b only slides the graph, so it does not change the period. Okay? Remember our puzzle? Sine x needs two pi to repeat, but mod sine x repeats after pi. Can you see why it needs only pi? Here is why. Our allied angle rule says sine of x plus pi equals minus sine x. The modulus removes that minus sign, and so does squaring; the little two up high means squared. Same for cos. Clear? For a sum of periodic functions, the period is usually the LCM, the least common multiple, of the separate periods. Notice the word usually; our puzzle comes back in example two. Right? And one more trap. The small circle up high is the degree sign. Sine x degrees equals sine of pi times x over one hundred and eighty, so its period is three hundred and sixty, not two pi. Got it?

4:27Example 1: simplify allied angle ratios to tan squared theta

Alright, example one: simplify this fraction. On top: sine of pi minus theta, times cos of pi over two plus theta, times tan of three pi over two minus theta. At the bottom: cos of two pi minus theta, times sine of three pi over two plus theta, times cot of pi plus theta. Pause the video and try it first. Done? Okay, let's check. Sine of pi minus theta: n is two, even, and sine is positive in the second quadrant, so sine theta. Cos of pi over two plus theta we already did: minus sine theta. Right? Now tan of three pi over two minus theta. Here n is three, odd, so tan becomes cot. Is it plus cot theta, or minus cot theta? It is plus cot theta, that is correct! The angle lies in the third quadrant, and there tan is greater than zero. Clear? Now the bottom. Cos of two pi minus theta: n is four, fourth quadrant, cos positive, so cos theta. Sine of three pi over two plus theta: n is three, so sine becomes cos, and sine is negative in the fourth quadrant, giving minus cos theta. And cot of pi plus theta is cot theta. Got it? Put it together. The two minus signs cancel, and cot theta cancels from top and bottom. Sine squared theta over cos squared theta is tan squared theta, wherever the original expression is defined. Six terms, and one neat answer!

6:02Example 2: five periods, including mod sin x plus mod cos x

Now a harder one. Find the periods of: part a, sine three x; part b, tan of two x over three; part c, sine x plus cos of x over two; part d, our puzzle, mod sine x plus mod cos x; and part e, sine of x over three, plus tan of x over two. Pause the video and try them first. Done? Okay, let's check. In part a, the number in front of x is three, so the period is two pi over three. In part b, careful: tan has period pi, not two pi. So pi over two thirds gives three pi over two. Right? In part c, cos of x over two has period two pi over one half, which is four pi, and the LCM with two pi is four pi. In part e, the periods are six pi and two pi, so we get six pi. Okay? And finally, our puzzle, part d. Each term has period pi, so the LCM is pi. But wait, could there be a smaller period? Yes, there is a smaller one. Replace x by x plus pi over two; that is what the arrow means. Then sine x becomes cos x, and cos x becomes minus sine x. After the modulus, the two terms just swap, so the sum does not change. With me? So the period is pi over two, smaller than each term's period, and nothing smaller works. That solves our puzzle! When the terms are even functions or powers, always check for a smaller period. Clear?

7:40Quick revision: allied angle rule and trig function periods

So, quick revision. Allied angles: if n is even, keep the function; if n is odd, switch sine with cos, and so on. The sign comes from the original function, in the angle's quadrant. Periods: two pi for sine and cos, pi for tan and cot, and f of, a times x, plus b, has period T over mod a. Now go and try the practice problems on this page. All the best!