PrepShorts · Study sheet · Class 10 Mathematics · Chapter 8, Introduction to Trigonometry
Chapter 8 · Introduction to Trigonometry
The extreme cases at 0° and 90°, and the ratios that stop being defined
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There is no right triangle with an angle of zero in it, and none with a second right angle. So the two columns at the ends of the table were never measured off anything - they were chosen. This is why the choice is forced, and why four cells have no value at all.
The idea
There is no right triangle with an angle of 0°, and none with a second angle of 90°, so the ratios at these two angles cannot be measured off a figure the way every earlier value was — they have to be defined. The chapter chooses those definitions by shrinking a real triangle until it collapses and taking the values the ratios were heading towards. That is why four cells of Table 8.1 read as undefined rather than as some number: at the moment of collapse a side has gone to zero, and the four ratios that carry that side underneath have nothing left to divide by. The gaps in the table are the honest record of a division that cannot be performed.
What you should be able to do
- Explain why 0° and 90° cannot appear as the working angle in an actual right triangle
- Describe what happens to each side of the triangle as the angle is squeezed towards 0°, and again as it is opened towards 90°
- State the four defined values the chapter adopts at these two angles, and say on what grounds
- Derive the remaining values at 0° and at 90° from those definitions
- Identify which four entries of Table 8.1 are undefined and give the reason in each case
- Read Table 8.1 as a whole and describe how sine and cosine behave across the range
- Judge true-or-false claims about how these ratios change as the angle grows
Words to know
| Term | Definition in one line | First introduced |
|---|---|---|
| not defined | the entry the table uses where the ratio would require dividing by a side that has vanished | printed in §8.3, p. 124, and in four cells of Table 8.1, p. 125 |
| Table 8.1 | the chapter's reference grid of all six ratios at five angles | printed as the table caption, §8.3, p. 125 |
| cosecant | the reciprocal of the sine of the same angle, and therefore undefined wherever the sine is zero | printed in §8.2, p. 115; its undefined case appears in §8.3, p. 124 |
| cotangent | the reciprocal of the tangent of the same angle | printed in §8.2, p. 115; its undefined case appears in §8.3, p. 124 |
| increases | the word the chapter's Remark uses for the direction the sine moves as the angle grows | printed in the Remark below Table 8.1, §8.3, p. 125 |
| degenerate case | an added name for the collapsed final panel of each strip, where the triangle has stopped being a triangle | an added term; the chapter draws this state twice and does not name it |
| limiting value | an added name for the number a ratio is heading towards as the triangle collapses | an added term; the chapter argues informally towards these values without naming the idea |
Where people slip up
- "Not defined means the answer is infinity." It means the operation cannot be carried out. Nothing has a value here — there is no number, not a very large one.
- "Not defined means zero." The two are opposites in this table: an entry is undefined precisely where its reciprocal partner is zero.
- "0° and 90° are just two more angles." Neither can be the working angle of a right triangle. Everything in this topic is definition supported by a picture, not measurement.
- "The triangle in the last panel is very thin." In the last panel of each strip there is no triangle at all. The chapter draws the collapse, and that is the point.
- "The cosine increases with the angle, like the sine." It falls. This is the single most-tested confusion in the section, and Exercise 8.2 question 4 asks it directly.
- "sin θ = cos θ, since they're both about the same triangle." They agree at 45° and part company everywhere else. One row rises while the other falls; they can only cross once.
- "You can memorise the table as a block of symbols." The four gaps are the part that has to be reasoned, and they are also the part that is asked about. Anchor each gap to the side that vanished.
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Worked answers: Exercise 8.1 · Exercise 8.2 · Exercise 8.3 · this video explains Exercise 8.2 Q1, Exercise 8.2 Q2, Exercise 8.2 Q4
Transcript2,047 words
Every value you have collected so far came off a figure. Now try that for an angle of zero degrees. Draw a right triangle. Put the right angle in the corner. Now make one of the other two angles zero. You cannot. The moment that angle reaches zero the side facing it has gone, and the three corners are sitting in a straight line. That is not a very thin triangle. It is not a triangle.
Now try ninety degrees. The right angle is already sitting in one corner, and the two remaining angles have exactly one right angle to share between them. You can see the sharing in the ratios. Read the same triangle from the other acute corner and the near and far sides swap over, so the sine at one corner is the cosine at the other. So the moment one of them reaches ninety, the other one is zero, and you are back to three points in a line.
And yet both of them sit at the ends of the table you are building, and both of them turn up in exercises. They are chosen. This whole video is about why the choice is not arbitrary. Start with a right triangle you can actually draw. Call the corner we are watching A, put the right angle at B, and let C sit above B. Keep the base fixed. Keep B where it is. And slide C down the vertical towards B.
Watch six frames of that. In each frame after it I will leave the opening triangle behind as a dotted outline, so you can see how far things have moved. Second frame: C has dropped. The angle at A has closed. The slanted side, the one running from A up to C, is settling down onto the base. And in the sixth frame C has arrived at B. The vertical side is gone entirely. What is left on the board is a single segment with two letters on top of each other at one end.
I want you to look at that last frame rather than skip past it. It is the frame that forces everything that follows. The side facing the angle at A shrank towards nothing. The side beside the angle never moved at all. And the slanted side, the longest one, closed the gap and ended up lying along the base. Now take those three facts to the ratios. The sine is the far side over the longest side.
Its top is heading for nothing while its bottom is heading for the length of the base, which is a perfectly ordinary number. The cosine is the near side over the longest side. Its top never moved, and its bottom is settling onto exactly that same length. A number over itself is one. Make the standing side one hundred times the vanishing one. Then the sine is already below one hundredth. Not near it. Below it.
Name any target you like, however small, and I can name a triangle in this family whose sine is under it. That is what heading towards nothing means, said in a way you could check. No triangle in that family ever GETS to nothing. Every one of them is a real triangle with a real angle, and its sine is strictly between nothing and one. So here is the decision.
There is no triangle at the end of that strip to read a value off. But everything approaching it is behaving in one particular way, and only one number sits at the end of each of those two roads. So we define the sine of zero degrees to be zero. And we define the cosine of zero degrees to be one. I want to be straight with you about that, because it is the one place in this whole topic where a value is put in by hand.
It is not arbitrary either. Any other choice would put a jump in the middle of something that was moving smoothly. Two values, chosen. Now watch how much they buy. The other four ratios at zero degrees are not choices. They follow. The tangent is the sine divided by the cosine. That is zero divided by one, which is zero. The secant is one divided by the cosine. That is one divided by one, which is one.
So four of the six are settled, and not one of the four needed a picture. Two chosen values went in, and the ordinary rules that connect these six ratios did the rest. Two entries left. The cosecant is one divided by the sine. The sine at zero degrees is zero. So the cosecant would be one divided by zero. There is no such number. Not a very large number. Not infinity. Nothing at all. The division cannot be carried out.
The cotangent is one divided by the tangent, and the tangent at zero degrees is also zero. Same story. Same reason. So we write not defined in both cells, and not defined means exactly one thing here: we tried to divide by a side that had vanished. The cosecant is the longest side over the far side, and the far side is gone. The cotangent is the near side over the far side, and the far side is gone.
Both of them were standing on the same leg, and that leg is what the shrinking took away. That is worth holding on to. A gap in this table is never a mystery. It always names a side that is not there. Now the other end, and it is the mirror image. This time keep C where it is, keep the right angle at B, and slide A along the base towards B.
As A comes in, the angle at A opens out. Five frames of that, with the earlier positions left dotted behind. Then A moves in, and the angle at A widens while the angle up at C closes. The slanted side is settling onto the vertical one. The base is shrinking towards nothing. And in the fifth frame A has reached B. The base is gone and the slanted side is lying flat against the vertical.
So this time it is the near side that vanished, not the far one. The sine is the far side over the longest side, and those two are settling onto each other. A number over itself is one. The cosine is the near side over the longest side, and the near side is the one going away. So the cosine is heading for nothing. And it is the same kind of statement as before. Make the standing side one hundred times the vanishing one and the cosine is already below one hundredth.
So we define the sine of ninety degrees to be one, and the cosine of ninety degrees to be zero. The cosecant is one over the sine, which is one over one, so one. The cotangent is the cosine over the sine, which is zero over one, so zero. The tangent is the sine over the cosine, and the cosine here is zero. No such number. The secant is one over the cosine. Same zero underneath. No such number.
Two more gaps, and again both of them are standing on the side that went away. Six rows, one for each ratio. Five columns, for zero, thirty, forty five, sixty and ninety degrees. Thirty cells. The middle three columns you already built out of two constructions. The two outer columns are what this video has just settled. The sine row reads zero, one half, one over root two, root three over two, one.
The cosine row reads one, root three over two, one over root two, one half, zero. The tangent row reads zero, one over root three, one, root three, and then not defined. The cosecant row starts with not defined, then two, root two, two over root three, one. The secant row reads one, two over root three, root two, two, and then not defined. And the cotangent row starts with not defined, then root three, one, one over root three, zero.
Four gaps in thirty cells. The cosecant row should be the sine row turned upside down, cell by cell. It is. The secant row should be the cosine row turned upside down. It is. And the cotangent row should be the tangent row turned upside down. It is. And every single gap sits directly above or below a zero. That is not a coincidence, it is the whole rule: a ratio is undefined exactly where the ratio it inverts is nothing.
The sine row: zero, one half, one over root two, root three over two, one. Every step is a step up. It starts at nothing and finishes at one. The cosine row: one, root three over two, one over root two, one half, zero. Every step is a step down. Same five numbers, walked backwards. So as the angle grows from nothing to a right angle, the sine climbs and the cosine falls.
The two rows cross. They agree at forty five degrees, and at none of the other four columns. One is going up and one is going down, so they can only meet once. Look at where the sine and the cosine live. Both of them are a leg divided by the longest side. The longest side is the biggest number in the triangle, so those two ratios can never get above one.
Across the whole table they stay between zero and one, ends included, and they never once go missing. The other four are not like that at all. Push the angle towards nothing and the cosecant sails past ten, then past a hundred, then past a thousand. Name a size and that family gets past it. But it never arrives anywhere, and at the end of the strip there is nothing at all.
That is exactly why those four are the ones with the gaps, and the sine and the cosine are the two you can always ask about. Four claims. Decide each one before I do. One. The sine of an angle grows as the angle grows. True. That is the top row of the table, climbing from nothing to one. Two. The cosine of an angle grows as the angle grows.
False, and this is the single most common slip in the topic. The cosine falls. Same numbers as the sine row, walked the other way. Three. The cotangent of zero degrees is not defined. True. The cotangent stands on the side that vanishes as the angle closes. Four. The sine and the cosine of an angle are equal for every angle. False. They are equal at forty five degrees, and at none of the other four columns of the table.
That is what the table is for. It is not a list to memorise, it is a picture of two quantities moving in opposite directions. One last question, because it uses both ends at once. For which angle is the sine of twice the angle equal to twice the sine of the angle? Try zero degrees. The sine of zero is zero, and twice zero is zero. Both sides are zero. It holds.
Try thirty. Twice thirty is sixty, and the sine of sixty is root three over two. Twice the sine of thirty is twice one half, which is one. Root three over two is not one, so thirty fails. Try forty five. Twice forty five is ninety, and the sine of ninety is one. Twice the sine of forty five is root two, which is not one. So forty five fails.
And that is the honest way to answer it: three of the candidates were settled by the board in front of you, and the answer is zero degrees. The two columns you cannot measure turned out to be the two that did the most work. You could not draw either of them. You watched a triangle collapse, you took the two values it was heading for, and everything else in both columns came out of ordinary arithmetic.
Including the four places where the arithmetic looked at what was underneath, found nothing there, and stopped.
Where this fits
Taken from the notes each video was made from, not from the reading order — these are the ideas this one rests on and the ones that later rest on it.
Builds on
- Defining sine, cosine and tangent, then their three reciprocalsClass 10 · Ch 8, Introduction to Trigonometry
- Squeezing 30°, 45° and 60° out of two special trianglesClass 10 · Ch 8, Introduction to Trigonometry
Comes up again in
- Dividing Pythagoras through by the hypotenuse squaredClass 10 · Ch 8, Introduction to Trigonometry
- Two more identities from the same equation, and the angles they hold forClass 10 · Ch 8, Introduction to Trigonometry