Part C: Concept Checks

Transcript

0:00Introduction

Hello everyone! Let me start with a quick puzzle. Which is larger, sine one or sine two, in radians? Some students say sine two, because two is bigger. Others say sine one, because sine falls after pi over two. So who is right? Hold that thought. Today we do seven quick concept checks from Part C of our trigonometry problem bank. Checks like these come up often in JEE, so let's begin.

0:28Does sin inverse of sin x always equal x?

Let us start with check number one. Sine inverse x has a small minus one up top; it means inverse, not a power. Its principal values lie from minus pi over two to pi over two; the curly e-shaped sign means belongs to, and the square brackets include both ends. Okay? One way round is always safe: sine of, sine inverse x, in round brackets, equals x, for every x from minus one to one. Right? Now the other way round. Is sine inverse of, sine x, equal to x for every real x? No, it is not, and here is why. Sine inverse only gives back angles from minus pi over two to pi over two. So sine inverse of, sine x, equals x only when x lies in that interval. The left graph is a sawtooth, a zig-zag, which follows the line y equals x only on the middle piece; the right one is the same idea for cos inverse of, cos x. With me? For example, sine pi is zero, and sine inverse of zero is zero, so sine inverse of, sine pi, is zero, not pi. On the left graph, at x equals pi, the zig-zag sits on the x axis. Clear?

1:42Is cos inverse of minus x equal to minus cos inverse x?

Check number two is about negative inputs. Sine inverse and tan inverse are odd: sine inverse of minus x is minus sine inverse x, and the same for tan inverse. Sine inverse on the left, and tan inverse, the solid curve on the right, are both symmetric about the origin; the dashed curve is cot inverse. Got it? So, is cos inverse of minus x also equal to minus cos inverse x? Oh no, careful, the answer is no! Look at the middle graph: cos inverse x lies from zero to pi, so it is never negative. But minus cos inverse x is never positive, and it is negative for every x less than one, so that rule cannot work. Okay? The correct rule is: cos inverse of minus x equals pi minus cos inverse x. Pi minus an angle from zero to pi stays from zero to pi, so it fits the range. Right?

2:37Period of |sin x| + |cos x|: pi over 2, not pi

Check number three is about periods. Mod sine x, with two vertical bars for the modulus, has period pi, and so does the modulus of cos x. With me? So what is the period of their sum? The LCM, the lowest common multiple, of pi and pi is pi. So the period is pi, isn't it? No, no, wait, check a smaller value first. Replace x by x plus pi over two; the arrow means changes to. Sine of, x plus pi over two, is cos x, and cos of, x plus pi over two, is minus sine x; the modulus removes the minus sign. So the two terms just swap places. Right? Swapping two terms does not change their sum, so the period is pi over two, not pi. When the terms are even functions or powers, test a smaller value before trusting the LCM rule.

3:31Equal ratios: must the angles be equal?

Clear? Now for check number four. If tan A equals tan B, must the angles be equal? Not necessarily, and the graph shows why: the tan graph on the right repeats every pi. The course's general solution says: tan theta equals tan alpha implies theta equals n pi plus alpha. Theta and alpha are Greek letters, the double arrow means implies, and n belongs to the double-struck Z, the integers. Right? So tan A equals tan B only tells us that the angle A equals n pi plus B, for some integer n; the angles are equal only when n is zero. Got it? Check number five is a close cousin. For squares, the course gives: sine squared theta equals sine squared alpha implies theta equals n pi plus or minus alpha; a plus over a minus is read plus or minus. Okay? So, is the general solution of sine squared theta equals one quarter, n pi plus or minus pi over six? Yes, this time it is true! Sine pi over six is one half, so one quarter is sine squared of pi over six, and theta equals n pi plus or minus pi over six. Not every check is a trap, so think before you say no. With me?

4:42Example 1: which is larger, sin 1 or sin 2?

Now Example one, our puzzle from the start. Which is larger, sine one or sine two? Pause the video and try it first. Done? Okay, let's check. Pi over two is about one point five seven; the wavy equals sign means approximately. So two radians lies beyond pi over two, where the solid sine curve in the figure has started to fall. Right? So does that make sine two smaller than sine one? Not so fast, let us bring two back before pi over two. Sine of pi minus theta equals sine theta, so sine two equals sine of, pi minus two, and pi minus two is about one point one four. Okay? Now pi minus two, about one point one four, lies between one and pi over two, where sine is increasing; the pointed signs mean less than and greater than. So sine of, pi minus two, is greater than sine one. Clear? So sine two is larger than sine one. The students who said sine two were right, but for the wrong reason. Got it?

5:48Example 2: tan inverse sum formula, when to add pi

Now Example two, a little harder. Is tan inverse x plus tan inverse y equal to tan inverse of, x plus y, over one minus x times y, whenever x times y is not equal to one? The slashed equals sign means not equal to. Pause the video and try it first. Done? Okay, let's check. The answer is no. The formula is true only when x times y is less than one. Right? With x and y both positive and x times y greater than one, we add pi; with both negative, we subtract pi. With me? Let us test it on tan inverse two plus tan inverse three, from the inverse functions chapter, where x times y is six, greater than one. What does the formula alone give? Two plus three is five, and one minus six is minus five, so we get tan inverse of minus one, which is minus pi over four. But two positive angles cannot add up to a negative angle, right? Yes, exactly, the formula alone misses a pi. Both numbers are positive with x times y greater than one, so add pi: pi minus pi over four gives three pi over four. Always check x times y first. Clear?

7:04Quick revision: all seven trigonometry concept checks

So, quick revision. Sine inverse of, sine x, equals x only from minus pi over two to pi over two, and cos inverse of minus x is pi minus cos inverse x. Modulus of sine x plus modulus of cos x has period pi over two, and tan A equals tan B gives the angle A equal to n pi plus B, for an integer n. Sine squared theta equals one quarter gives n pi plus or minus pi over six, and sine two is larger than sine one. And the tan inverse sum formula needs x times y less than one. Well done on all seven checks! Three more are waiting on this page, so go and try them. All the best!