A Good Question

If Tan x = 3/4 what does sin x =?So half my Year Eleven’s are missing, out on a Geography field trip. No point starting anything new, time to whip out some trusty revision material – “20 Homework Quickies” if I remember correctly. 20 short questions on a range of GCSE topics: 3x + 4 = 5 x – 2,  write 0.00012345 in Standard Form etc.

All good stuff, helping the pupils re-visit topics they may not have seen for many months, and telling me what areas they have collectively forgotten about.

But question number 4 was a real gem. Question number 4 was:

If tan x = 3/4 what is sin x?

A seemingly innocuous little question, reach for the calculator, type in the numbers and out pops the answer.

STOP! Put down the calculator. Let’s think about the question:

“When do we use tan?” Cue inevitable jokes about sun tan and fake tan.

“When do we use tan?”


“When do we use tan?” Beginning to sound like a broken record, but I’m not giving up.

“Right angled triangles”

“Excellent.  And what else do we do with right angled triangles and tan?”

“Sin” (to rhyme with gin.)

“We pronounce it ‘sign.’ And what else?”


“Oooh, I went there for my holidays, and got a nice tan” says the class comedian. Never.

“So we’ve got a right angled triangle, Tan, Sine and Cos.  What do we do next?”

“Sohcahtoa” comes a random cry from the back of the class.

“Label the sides. What are we going to call them?”


“O, A and H”

“Brilliant. And which is which?” (At this point you can have a little diversion into A level Statistics. How many different ways can you label the three sides of a triangle with O, A & H? The probability is quite high that your class with have found them all before you can work out how many there are.)

You agree on which of the six different permutations is correct.


You begin to think that there is an echo in the classroom.

“So what is the purpose of ‘SOHCAHTOA’, other than a phrase to be shouted out as you know it has something to do with triangles?”

” Sin equals O over H, Cos equals A over H, Tan equals O over A”

You console yourself with the thought that a least one pupil will pass this summer.

“And what else helps us with right angled triangles?”


“The angles in a triangle add up to 1800


“But what – or who – else helps us when solving right angled triangle problems? Think of that Greek bloke.”


“Excellent. What is Pythagoras Theorem?”

“a2 + b2 = c2

“Brilliant. So can we work out the hypotenuse?”

“If its the hypotenuse, why do we call it c?”

“Just work it out”


“And sin x is …?”

“O over H”

“And O is what? H is what? So, the answer to our question, what is sin x is …”

We get there in the end. Such a simple little question, but so much maths came out of it.

I bet they didn’t have that much fun counting cars on the geography field trip.

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