Definitions and Principal Values

Transcript

0:00Introduction

Hello everyone! Which angle has sine equal to one half? Pi over six, you will say. But sine of five pi over six is also one half, and so is sine of thirteen pi over six. Infinitely many angles! So when we write sine inverse of one half, which angle do we mean? That is today's topic, principal values, and questions on them come up often in JEE papers. So, let's begin.

0:28First one-one, then the inverse

So, recall one fact from functions. Only a one-one function has an inverse; one-one means different inputs give different outputs. Okay? Now let me ask you this. Is sine x one-one on the double-lined capital R, the set of all real numbers? No, it is not, and our puzzle shows why. Sine of pi over six and sine of five pi over six are both one half; pi here is the Greek letter. Two angles give the same output. Right? So we keep one piece, from minus pi over two to pi over two. The square brackets mean both ends are included. On this piece, sine is one-one and takes every value from minus one to one, so it is onto. With me? Its inverse there is sine inverse x. The small minus one up top is not a power; it means inverse. Some books write arc sine x, the same thing. Clear? So, when y equals sine inverse x, y is the angle in this interval whose sine is x. It is always an angle. Got it? So our puzzle is solved. Of all those angles, only pi over six lies in this interval, so sine inverse of one half is just pi over six. Okay?

1:46A very common trap

Now a quick question. Is sine inverse x the same as one over sine x? No, no, careful, this is a very common mistake! The equals sign with a slash means not equal to. Sine inverse x is an angle, not one over sine x. Right? One over sine x is cosec x. To write it as a power, put sine x in round brackets, with the minus one outside. Those brackets make all the difference. With me?

2:18Principal value branches

Now, the other five are defined the same way, on a piece where the function is one-one. Sine inverse x has domain minus one to one, and its range, minus pi over two to pi over two, holds its principal values. It is the dark blue curve, okay? Now let us look at cos inverse. Can cos use the same piece, from minus pi over two to pi over two? No, it cannot. Cos of an angle equals cos of minus that angle, so cos is not one-one there. So cos inverse uses zero to pi, where cos falls from one to minus one: the orange curve. Right? Tan inverse x accepts every real number, so its domain is capital R, all real numbers. Its range has round brackets, so the ends, minus pi over two and pi over two, are left out, since tan is not defined there. It is the solid green curve, which never reaches the dashed lines. Clear? Cot inverse x also accepts every real number. Its range is zero to pi, ends left out, since cot is not defined where sine is zero. It is the dashed blue curve. Got it? Look at the shapes. Sine inverse and tan inverse always rise, so they are increasing, and they are odd, symmetric about the origin. Okay? But cos inverse and cot inverse always fall, so they are decreasing, and they are neither odd nor even. Keep this in mind. With me?

3:45Sec and cosec inverse: watch the gaps

Now for the last two functions. Sec inverse x takes inputs from minus infinity to minus one, union one to infinity; the sideways eight is infinity, and the cup sign is union. Its range is zero to pi, with pi over two removed; the minus sign here means set minus, and pi over two sits in curly brackets. So why remove pi over two? Because cos of pi over two is zero, so sec of pi over two is not defined. Cosec inverse x has the same domain, and its range is minus pi over two to pi over two, leaving out zero, where cosec is not defined. Clear?

4:26Example 1: six principal values and the cot inverse trap

Alright, here is our first example. Find the principal values of sine inverse of minus one half, cos inverse of minus root three over two, tan inverse of minus one, cot inverse of minus root three, sec inverse of minus two, and cosec inverse of minus root two. Root is the square root sign. Pause the video and try them first. Done? Okay, let's check. Every time, find the angle in the principal range with that value. Sine inverse of minus one half is minus pi over six. For cos inverse of minus root three over two, the angle is from zero to pi and cos is negative, so it is pi minus pi over six, which is five pi over six. Okay? Tan inverse of minus one is simply minus pi over four, inside its open range. Right? Now cot inverse of minus root three. Cot of minus pi over six is minus root three, so the answer is minus pi over six, isn't it? No, no, careful, that is the trap! Minus pi over six is not in the range of cot inverse, which is zero to pi. There, cot is negative after pi over two, so the answer is pi minus pi over six, that is, five pi over six. Clear? Sec inverse of minus two means the angle whose cos is minus one half, taken from zero to pi. That angle is two pi over three. Got it? And cosec inverse of minus root two is the angle from minus pi over two to pi over two whose sine is minus one over root two. That is minus pi over four. With me?

6:06Example 2: solve an inverse trig equation from domains only

Now here is a harder one. Solve this: tan inverse of root of x squared plus x, plus sine inverse of root of x squared plus x plus one, equals pi over two. Pause the video and try it. Here is a hint: check the domains first. Done? Okay, let's check. Sine inverse accepts inputs only from minus one to one. So what must be true of the root of x squared plus x plus one? Yes, exactly, it must be at most one; this sign means less than or equal to. Minus one is no problem, since a square root is never negative. Squaring, x squared plus x plus one is at most one, so x squared plus x is at most zero. Okay? Now look at the first term. Tan inverse accepts any real number, but the square root inside it needs x squared plus x to be at least zero; this sign means greater than or equal to. Right? At most zero, and at least zero, so x squared plus x is exactly zero! Factor it: x times, x plus one, equals zero, so x is zero or minus one. Clear? Let us check both. For each value, x squared plus x is zero, so the left side is tan inverse of zero, plus sine inverse of one, which is zero plus pi over two. Both values work! That is a topper's habit: with inverse functions, check the domains first. Got it?

7:43Quick revision: principal ranges and the 1 over sin x trap

So, let us do a quick revision. Sine inverse x is the angle from minus pi over two to pi over two whose sine is x, not one over sine x. Cos inverse gives zero to pi, tan inverse gives minus pi over two to pi over two with the ends left out, and cot inverse gives zero to pi, ends left out. Pick the answer inside the principal range, and check domains first. Now try the practice problems on this page. All the best!