Mixed Problems: Set 3 (Inverse Functions and Triangles)

Transcript

0:00Introduction

Hello everyone! Let us start with a small puzzle. Tan inverse undoes tan, right? So tan inverse of tan four should simply be four. But tan inverse only gives answers between minus pi over two and pi over two, and pi over two is about one point five seven. So what is the real answer? Example one settles it. This is Set three of our mixed JEE Main pattern problems: inverse functions and triangles, which come up often. Let's begin.

0:31Set 3: what it holds

Set three holds fifteen solved problems: first inverse functions, then triangles and heights, at one to two minutes per problem. Let us sharpen three tools from Chapters four and five, and then solve two problems together. With me?

0:46Tool 1: principal value branches

Tool number one, from Chapter four: the principal value branches. Sine inverse x, written with a small minus one up high, is not one over sine x, okay? It is the angle whose sine is x. The rounded symbol like an e means belongs to: x lies from minus one to one, and the answer from minus pi over two to pi over two. Square brackets include both end points, and the dark blue graph on the left shows this. Clear? Cos inverse x takes the same inputs, minus one to one, but its answer lies from zero to pi. That is the orange curve in the middle graph, falling from pi down to zero. Got it? Now tan inverse x, where x can be any real number. Look at the green curve in the last graph, and the dashed lines at minus pi over two and pi over two. Can the curve ever reach them? It never reaches them, that is correct! So tan inverse x lies strictly between minus pi over two and pi over two. The round brackets mean the end points are left out, and the double struck capital R means all real numbers. The dashed blue curve is cot inverse x, which we do not need today. Okay?

2:02Tool 2: when does the inverse undo the function?

Tool number two: does the inverse really undo the function? One way round, always: sine of sine inverse x is x, for every x from minus one to one. Right? Now the other way round. Is sine inverse of sine x always equal to x? Not always, and here students lose marks. Sine inverse only answers from minus pi over two to pi over two, so sine inverse of sine x equals x only when x itself lies there. With me? The same for tan: tan inverse of tan x equals x only when x lies strictly between minus pi over two and pi over two. Okay? So what if x is outside? The period of tan is pi, so tan of x minus pi equals tan x. Shift x by pi until it lands inside the branch; that is your answer. Clear?

2:56Tool 3: the sine rule

Tool number three, from Chapter five: the sine rule. Capital A, B, C are the angles, and small a, b, c are the sides opposite them. Small a over sine capital A, equals small b over sine capital B, equals small c over sine capital C, and each equals two times R, where R, the orange line from the centre O, is the circumradius, the radius of the light grey-blue circle through all three corners. The green incircle, with centre I and radius r, is not needed today. Okay? So, with two sides and one angle, the sine rule gives the sine of another angle: sine capital C equals c over b, times sine capital B, right? But here is a question for you. Inside a triangle, does knowing sine capital C fix the angle capital C completely? It does not, and that is the trap! By allied angles, sine of one eighty degrees minus capital C equals sine capital C, okay? The little circle up high means degrees. Both angles have the same sine, so we get two choices. Got it? So how do we pick one? The smaller side faces the smaller angle: if small c is less than small b, then capital C is less than capital B, okay? The arrow with a double line means implies. Clear?

4:21Example 1: tan inverse of tan 4, domain of sin inverse log

Now, remember our puzzle? Here it is as Example one, marked easy. Find tan inverse of tan four, and the domain of sine inverse of log, to the base three, of x over three. Pause the video and try it first. Done? Okay, let's check. Many students write four straight away. Is that right? No, and tool two tells us why. Pi over two is about one point five seven, and three pi over two is about four point seven one; the wavy equals sign means approximately. So four lies between pi over two and three pi over two, outside the branch of tan inverse. Okay? So we use the period. Four minus pi is about zero point eight six, and that lies between minus pi over two and pi over two. And tan of four minus pi equals tan four, because tan repeats every pi. Right? So tan inverse of tan four equals four minus pi, the course answer. Puzzle solved: four was never possible. Clear? Now the second part, the domain. Remember tool one: what inputs does sine inverse accept? From minus one to one, exactly! So log, to the base three, of x over three lies from minus one to one; the sign with a bar under it means less than or equal to. Undo the log: three to the power minus one is one third, and three to the power one is three. The base is bigger than one, so the signs keep their direction; the double arrow means if and only if. With me? Multiply through by three, and x lies from one to nine. Every such x is positive, so the log is defined as well. So the domain is the closed interval from one to nine, the course answer. Got it?

6:14Example 2: angles in AP, b:c = root 3 : root 2, find A

Now Example two, marked medium. The angles of a triangle are in arithmetic progression, A P on screen, and small b is to small c as root three is to root two; the colon means ratio. Find the angle capital A, okay? Pause the video and try it first. Done? Okay, let's check. Capital A, B, C in arithmetic progression means capital A plus capital C equals two times capital B, right? And the three angles add up to one hundred and eighty degrees, so three times capital B is one hundred and eighty, and capital B is sixty degrees. Okay? Now tool three, the sine rule. Sine capital C equals c over b, times sine sixty degrees. That is root two over root three, times root three over two; the dot means times. The root threes cancel, and we get one over root two. Right? Now, is capital C forty five degrees, or one hundred and thirty five degrees? Forty five degrees, yes, exactly! Small c is less than small b, so capital C must be less than capital B, which is sixty degrees. So one hundred and thirty five degrees is ruled out by the side check. Clear? So capital A equals one hundred and eighty, minus sixty, minus forty five, which gives seventy five degrees, the course answer. Got it?

7:43Quick revision: principal branches, shifting, sine rule

So, quick revision. First, the branches: sine inverse from minus pi over two to pi over two, cos inverse from zero to pi, tan inverse strictly between minus pi over two and pi over two. Second, tan inverse of tan x is x only inside the branch; otherwise, shift x by a multiple of pi, until it lands inside. Third, the sine rule gives sine capital C, and the smaller side faces the smaller angle. Now try the practice problems on this page, and then the rest of Set three. All the best!