The Six Trigonometric Ratios

Transcript

0:00Introduction

Hello everyone! In class ten, sine was opposite over hypotenuse, in a right triangle. Now, can you draw a right triangle with an angle of two hundred and ten degrees? No, because its angles add up to only one hundred and eighty degrees. And yet sine of two hundred and ten degrees exists, and it is negative! How can a ratio of lengths be negative? These six ratios come up often in JEE, and all of trigonometry builds on them, so let's find out.

0:32Six ratios of any angle

Take an angle theta, the Greek letter, measured from the positive x axis; it ends on its terminal ray. Pick any point P on that ray, with coordinates x and y in brackets, and let r be the distance O P, always greater than zero. And look, x and y can be negative! Okay? Sine theta equals y over r, and cos theta equals x over r. For an acute angle, these are just opposite over hypotenuse and adjacent over hypotenuse. On the unit circle in the figure, r is one, so P has coordinates cos theta and sine theta. The labels in the four corners are about signs, and we will come to them soon. Right? Now let me ask you this. In class ten, tan was opposite over adjacent. So, in terms of x and y, what should tan theta be? Yes, exactly, y over x! And cot theta is the other way round, x over y. So cot is simply one over tan. With me? Cosec theta equals r over y, which is one over sine theta, and sec theta equals r over x, which is one over cos theta.

1:44Values to know by heart

Clear? Now the values to know by heart. The top row gives each angle in radians and in degrees. Pi, the Greek letter, is about three point one four, and pi radians equal one hundred and eighty degrees, so pi over six is thirty degrees, and the small circle means degrees. So the five standard angles are zero, thirty, forty five, sixty and ninety degrees. Okay? The sine row climbs from zero to one: zero, one half, one over root two, root three over two, and one. The tick-shaped sign is the square root. Now, can you guess the cos row before it appears? Yes, exactly, it is the sine row in reverse! Now, tan theta is y over x. So what is tan of pi over two? At pi over two, x is zero, and we cannot divide by zero, so tan is not defined there. The rest of the row is zero, one over root three, one, and root three. Right?

2:43Signs: All Silver Tea Cups

Now the signs. Since r is always positive, every sign comes from x and y. The Roman numerals one to four in the figure are the quadrants. In quadrant one, x and y are both positive, so all six ratios are positive, with the plus sign. Clear? Now quadrant two, the top left of the figure. Here x is negative, but y is positive, and the figure says only sine and cosec are positive there. Can you tell me why just these two? Yes, exactly! Sine theta and cosec theta use only y and r, and both are positive here. The other four contain x, so they are negative. Okay? In quadrant three, x and y are both negative, so y over x is positive, and only tan theta and cot theta are positive. And two hundred and ten degrees lies in quadrant three, so its sine is negative. That solves our puzzle. With me? In quadrant four, x is positive and y is negative, so only cos theta and sec theta are positive. From quadrant one to four, remember All Silver Tea Cups: all, sine, tan, cos. Got it?

3:54Ranges, and where a ratio is not defined

Next, the ranges. In size, x and y never exceed r, so sine theta and cos theta lie between minus one and one. This curved symbol means belongs to, and the square brackets include both ends. Okay? Tan theta and cot theta can be any real number; the double-struck R is the set of all real numbers. Right? Now let me ask you this. Can sec theta ever be equal to one half? No, never, because sec theta is one over cos theta, which is at most one in size. So sec theta, and cosec theta too, is at least one or at most minus one. The infinity sign means no end, a round bracket leaves that end out, and the cup means union. Clear? Each ratio needs a denominator that is not zero. Tan theta and sec theta divide by x, so they fail where cos theta is zero, at two n plus one, times pi over two. Cot theta and cosec theta fail at n pi, where sine theta is zero. Here n is in the double-struck Z, the set of integers. Got it?

4:59Trick: one ratio and the quadrant give all six

Now a time-saving trick. Given one ratio and the quadrant, forget the sign, draw a right triangle with the given sides, and find the third side by Pythagoras. With me? Then read off the six ratios, and attach each sign from the quadrant: the triangle gives the size, and the quadrant gives the sign. Okay?

5:19Example 1: evaluate an expression at 30, 45, 60, 90 degrees

Now example one. On top, tan squared sixty degrees, plus four times cos squared forty five degrees, plus three times sec squared thirty degrees, plus five times cos squared ninety degrees. The small two means the whole ratio is squared. At the bottom, cosec thirty degrees, plus sec sixty degrees, minus cot squared thirty degrees. Pause the video and try it first. Done? Okay, let's check the top. Tan sixty degrees is root three, so its square is three. Cos forty five degrees is one over root two, so four times one half is two. Sec thirty degrees is two over root three, so three times four thirds is four. Cos ninety degrees is zero, so the top is nine. Right? Now the bottom. Cosec is one over sine, sec is one over cos, and cot is one over tan. So, what is cot thirty degrees? Yes, exactly, root three, since tan thirty degrees is one over root three. And cosec thirty degrees is one over one half, which is two, and sec sixty degrees is also two. So the bottom is two plus two minus three, just one. Clear? So the value is nine over one, which is just nine. It is easy once you know the table, okay?

6:41Example 2: sin theta = -3/5 in Quadrant III, find the rest

Now example two, a little harder. Sine theta is minus three over five, and theta lies in the third quadrant. Find the other five ratios, and then sec theta plus tan theta, over cosec theta plus cot theta. Pause the video and try it first. Done? Okay, let's check. Forget the sign for now, and draw a right triangle with opposite side three and hypotenuse five. By Pythagoras, nine plus sixteen is twenty five, so the third side is four. Got it? So cos theta is adjacent over hypotenuse, which is four over five, right? No, no, wait, that is the trap! Theta is in the third quadrant, where only tan and cot are positive. So cos theta is minus four over five, while tan theta is three over four and cot theta is four over three. With me? Sec and cosec, the reciprocals, keep the signs of cos and sine, so sec theta is minus five over four, and cosec theta is minus five over three. Okay? On top, minus five over four plus three over four gives minus one half. At the bottom, minus five over three plus four over three gives minus one third. Minus over minus is plus, and one half over one third gives three over two. Clear?

8:01Quick revision: six ratios, quadrant signs, one-ratio trick

So, quick revision. Sine theta is y over r, cos theta is x over r, tan theta is y over x, and the other three are their reciprocals. For the signs, All Silver Tea Cups, quadrant one to four. And with one ratio and its quadrant, the triangle gives the size, the quadrant the sign. Now try the practice problems on this page. All the best!