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The trig functions are about multiplication
When I was taught trigonometry for the first time, I learned it as ratios of sides of right-angled triangles.
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When will I ever use this?
"When will I ever use this?" is possibly a maths teacher's most feared student question. It conjures up all sorts of unpleasant feelings: anger that students don't see the wonder of the maths itself, sadness that they've come to expect maths is only worthwhile if it's usable for something, fear that if we don't respond right the students will lose faith in us, shame that we don't actually know any applications of the maths, but mostly just a rising anxiety that we have to come up with a response to it right now.
Really working together
Yesterday, I had one of those experiences in the MLC that makes me love my job.
A constant multiplied on will stay there
One of the most fundamental properties of the integral is that multiplying by a constant before doing the integral is the same as doing the integral and then multiplying by a constant. However, the way it's presented here makes it look like a rule for algebraic manipulation – I can move a constant multiple in and out of the integral sign. I do actually use it this way when I want to do algebraic manipulation – it comes in handy when I'm creating a reduction formula, for example. But most of the time when I do an integral, I don't use it that way at all.
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