Overview

Transcript

0:00Introduction

Hello everyone! Let us start with a small puzzle. Take five sine x plus twelve cos x. How large can it get? Sine x and cos x are each at most one, so five plus twelve gives seventeen, right? Hmm, we will check that today. This chapter has mixed problems in the JEE Main style, from all of trigonometry. So let us see what it holds, and settle our puzzle on the way. Let's begin!

0:28What this chapter holds

So, why does this chapter matter? Trigonometry often gives two or three questions in a JEE Main shift. So tell me, which kinds of question would you expect? Let us see how many you guessed. First, an identity, a value, or a bound, like our puzzle. That is Set one. Right? Second, an equation: count its solutions in an interval, or add them up. That is Set two. And third, an inverse function to simplify, or its domain, and often a triangle, or heights and distances. That is Set three. With me? These problems are original, one to two minutes each. The real past questions, from twenty twenty one to twenty twenty six, are at the end of Chapters one to five. Clear?

1:18Tools for Set 1: identities and bounds

Now, one tool for each type. First, a rule for every identity: it holds only where both sides are defined. Okay? So tan x and sec x need cos x not equal to zero; the equals sign with a slash means not equal to. And cot x and cosec x need sine x not zero. Right? Now the bound. Theta, the Greek letter, is the angle, and small a, small b are fixed numbers. Small a cos theta plus small b sine theta equals the root of small a squared plus small b squared, times cos of theta minus phi. The tick-shaped sign is the square root, the small two up top means squared, and phi, another Greek letter, has tan phi equal to small b over small a. With me? Now tell me, the cos of any angle lies between which two numbers? Between minus one and one, yes, exactly! So the expression lies between minus that root and plus that root; the sign with a line under it means less than or equal to. Got it?

2:24Tools for Set 2: equations

Next, equations. The functions repeat, so we list all the solutions with an integer n. Alpha and pi are Greek letters, and alpha is any one solution. If sine theta equals sine alpha, theta equals n pi, plus, minus one in brackets to the power n, times alpha. Here n belongs to the double-lined capital Z, the integers; the curved E sign means belongs to. Okay? If cos theta equals cos alpha, then theta equals two n pi plus or minus alpha. And if tan theta equals tan alpha, then theta equals n pi plus alpha. Keep these three in your notebook. Now, two warnings that cost marks. First, reject every value where a term is undefined, like tan theta at theta equal to pi over two, where cos theta is zero. Clear? Second, do not cancel a factor that can be zero, or you may lose roots. And if you square, check every root, since squaring can add extra ones.

3:28Tools for Set 3: inverse functions and triangles

Got it? Now inverse functions; the small minus one up top means inverse. Sine inverse x lies from minus pi over two to pi over two, and cos inverse x from zero to pi; square brackets include both ends. Tan inverse x lies strictly between minus pi over two and pi over two; round brackets leave the ends out. These are the principal values. Okay? Then, tan inverse x plus tan inverse y equals tan inverse of x plus y, over one minus x y, but only when x y is less than one; the sign pointing left means less than. Right? If x y is greater than one, where the sign points right, add pi when x and y are both positive, and subtract pi when both are negative. This extra pi is easy to forget. With me? For a triangle, capital A, B, C are the angles, and small a, b, c the opposite sides. Sine rule: small a over sine capital A, and so on, all equal two R, where R is the circumradius. Cosine rule: cos capital A equals small b squared plus small c squared minus small a squared, over two small b small c. Clear? And in heights and distances, the angles of elevation and depression are always measured from the horizontal. Draw the figure first. Got it?

4:57Example 1: integers in the range of 5 sin x + 12 cos x

Now, Example one, from Set one, and it is our puzzle in a new form. How many integers lie in the range of five sine x plus twelve cos x? Pause the video and try it first. Done? Okay, let's check. First, our quick answer: five plus twelve gives seventeen. For that, sine x and cos x must both be one, at the same x. Is that possible? No, that is never possible, and here is why. Sine squared x plus cos squared x is always one, so when sine x is one, cos x is zero. Right? Write it as twelve cos x plus five sine x, so small a is twelve and small b is five. The root of one hundred and forty four plus twenty five is the root of one hundred and sixty nine, which is thirteen. Okay? So it lies from minus thirteen to thirteen, taking every value in between. Now count the integers. Thirteen positive and thirteen negative make twenty six, right? Oh no, careful, you forgot zero, and that is the trap here! The integers run from minus thirteen to minus one, then zero, then one to thirteen. Clear? So the count is thirteen, plus one, plus thirteen, which is twenty seven. And our puzzle is settled: the greatest value is thirteen, not seventeen. Got it?

6:26Example 2: solutions of tan x + tan 2x = tan 3x in (0, pi)

Now a harder one, from Set two. Find the number of solutions of tan x plus tan two x equals tan three x, between zero and pi; the round brackets leave both ends out. Pause the video and try it first. Done? Okay, let's check. Three x is x plus two x, so tan three x equals tan x plus tan two x, over one minus tan x tan two x. Okay? Multiply across. Tan three x, times one minus tan x tan two x, equals tan x plus tan two x, and the equation says this sum is tan three x. Right? Open the bracket, and subtract tan three x from both sides. We get tan x times tan two x times tan three x equals zero; the double arrow means implies. We subtracted; we did not cancel. With me? Now each factor. From zero to pi, tan x is zero only at the two ends, zero and pi, and both are left out. Next, tan two x equals zero gives x equal to pi over two. So, do we keep it? No, no, wait, this is exactly the trap! At pi over two, cos x is zero, so tan x is not defined there. So we reject it. Clear? Last, tan three x equals zero gives pi over three and two pi over three, and every term is defined at both. So the number of solutions is two, not three. Got it?

8:03Quick revision: bounds, equations, inverse functions

So, quick revision. For bounds, small a cos theta plus small b sine theta lies between minus and plus the root of small a squared plus small b squared. For equations, use the general solution with an integer n, and reject undefined values. For inverse functions, use the principal values, and the tan inverse sum formula needs x y less than one. Now open Set one, on the next page, and start solving. All the best!