Properties and Compositions

Transcript

0:00Introduction

Hello everyone! Let me start with a quick puzzle. What is sine inverse of sine ten? Sine inverse cancels sine, so the answer is simply ten, right? Hmm, but wait. Sine inverse always gives an angle between minus pi over two and pi over two, roughly minus one point five seven to one point five seven. So the answer can never be ten! Today we find out when an inverse really cancels. This comes up often in JEE, so let's begin.

0:31Negative inputs: two different rules

So, first, let us see what happens to a negative input. Sine inverse of minus x equals minus sine inverse x, and the same holds for tan inverse and cosec inverse. The small minus one up top means inverse, not a power, okay? Here the minus sign simply comes outside. Now, does cos inverse follow the same rule? Let me ask you this: is cos inverse of minus x equal to minus cos inverse x? No, it is not, and this is a classic trap! Cos inverse gives an angle from zero to pi, the Greek letter pi, so it is never negative. The correct rule is: cos inverse of minus x equals pi minus cos inverse x. Right? Cot inverse and sec inverse also take values between zero and pi, so the same rule applies. Cot inverse of minus x equals pi minus cot inverse x, and the same for sec inverse.

1:31Complementary pairs and reciprocals

Clear? Next, the complementary pairs. Sine inverse x plus cos inverse x equals pi over two, and the same for tan inverse plus cot inverse, and for sec inverse plus cosec inverse. Each holds wherever both inverses are defined. Okay? Now the reciprocals. Cosec inverse x equals sine inverse of one over x, and sec inverse x equals cos inverse of one over x. With me? Similarly, cot inverse x equals tan inverse of one over x, when x is greater than zero; the sign between x and zero is the greater-than sign. Got it? But is this true for every x? Tell me, what happens when x is negative? Then tan inverse of one over x is negative, but cot inverse lies between zero and pi. So we add pi: for x less than zero, cot inverse x equals pi plus tan inverse of one over x. Students often forget this. Right?

2:34Which order cancels?

Now the big question from our puzzle. Below, you can see two zig-zag graphs that will help us. First, in this order, sine of sine inverse x equals x, for every x in the closed interval from minus one to one. The curly e-shaped sign means belongs to, and square brackets include the end points. Okay? Now flip the order. Is sine inverse of sine x equal to x for every x? Not always, and this is the key point! Sine inverse only gives angles from minus pi over two to pi over two, and the left graph below, sine inverse of sine x, never goes above pi over two or below minus pi over two. So sine inverse of sine x equals x only when x lies in that interval. Clear? In the same way, cos inverse of cos x equals x only for x from zero to pi, the range of cos inverse. Right? And tan inverse of tan x equals x only on the open interval, minus pi over two to pi over two; round brackets leave out the end points. The right graph is cos inverse of cos x, which stays between zero and pi. Both graphs zig-zag like the teeth of a saw, and on each straight piece, the value is plus or minus x, plus a multiple of pi. With me?

3:57Trick: bring the angle back

So how do we find sine inverse of sine theta? Theta, the Greek letter, is the given angle. The answer is the angle phi, another Greek letter, in minus pi over two to pi over two, with the same sine as theta. The same sawtooth graphs stay below, so you can see that range. Okay? Now tell me, how can we change an angle but keep its sine the same? We have two tools for that, exactly! Sine of pi minus theta equals sine theta, and adding or subtracting two pi, the period, changes nothing. Got it? For cos inverse of cos theta, use cos of two pi minus theta equals cos theta, with the period, and aim for zero to pi. Clear?

4:42Trap: conversions with a square root

One more trap before the examples. Converting one inverse into another with a square root is safe only when both angles lie in the same interval. So always check the sign of x. Okay? For x between zero and one, sine inverse x equals cos inverse of the square root of one minus x squared; the tick-shaped sign is the square root. Right? But for x between minus one and zero, sine inverse x is negative, while cos inverse never is. So now sine inverse x equals minus cos inverse of the root of one minus x squared. With me?

5:18Example 1: sin inverse sin 3pi/5, cos inverse cos 7pi/6

Now, the first example. Find sine inverse of sine three pi over five, cos inverse of cos seven pi over six, and tan inverse of tan three pi over four. Pause the video and try it first. Done? Okay, let's check. For the first one, sine and sine inverse cancel, so the answer is simply three pi over five, isn't it? No, no, that is exactly the trap! Three pi over five is more than pi over two, outside the range. Pi minus three pi over five is two pi over five, with the same sine, and it is in range. So the answer is two pi over five. Right? Next, seven pi over six is more than pi. Two pi minus seven pi over six is five pi over six, with the same cos, and it lies in zero to pi. So the second answer is five pi over six. Clear? For the third, three pi over four is outside minus pi over two to pi over two, but tan repeats every pi. Three pi over four minus pi is minus pi over four, which is in range. So the third answer is minus pi over four. Got it?

6:32Example 2: sin inverse sin 10 = 3pi - 10, cos 10 and tan 2

Now the harder one, and it is our puzzle from the start! Find sine inverse of sine ten, cos inverse of cos ten, and tan inverse of tan two, all in radians. Pause the video and try it first. Done? Okay, let's check. Pi is about three point one four; the wavy equals sign means approximately. Ten minus two pi, one period less, is about three point seven two, with the same sine. But that is still bigger than pi over two, about one point five seven. Okay? So, which tool brings ten minus two pi into the range? Sine of pi minus the angle, yes, exactly! Pi minus that bracket gives three pi minus ten, about minus zero point five eight, which is in range. So sine inverse of sine ten equals three pi minus ten, and our puzzle is solved. Right? Cos inverse of cos ten is similar. Two pi minus that same bracket gives four pi minus ten, about two point five seven, which lies in zero to pi. So cos inverse of cos ten equals four pi minus ten. With me? Finally, two is bigger than pi over two. Tan repeats every pi, so tan of two minus pi equals tan two, and two minus pi, about minus one point one four, is in range. So tan inverse of tan two equals two minus pi. Clear?

8:02Quick revision: minus x rules, pi/2 pairs, bring angle back

So, quick revision. For minus x, sine inverse, tan inverse and cosec inverse take the minus outside; cos inverse, cot inverse and sec inverse give pi minus. The pairs add to pi over two, and for negative x, cot inverse x is pi plus tan inverse of one over x. And sine inverse of sine x is x only in the principal range; otherwise, bring the angle back. Now try the practice problems on this page. All the best!