## All you need is sine

Today I was going through an M1 question with a year 13 student and was surprised to see the method he had used. The question involved finding an angle in a right angled triangle given the opposite and adjacent sides. The learner had used Pythagoras’s Theorem to find the hypotenuse then used the sine ratio to find the angle.

Puzzled I questioned further, thinking he may have instinctively found the hypotenuse without fully reading the question then having all 3 sides so going with the first. This turned out not to be the case:

“I know sine equals opposite over hypotenuse innit sir, I have trouble remembering the other ones so I just always use sine.”

This was extra interesting as earlier I had come across a markscheme which suggested the way to resolve a force at an angle of 30 degrees was to use Fsin30 for the vertical and Fsin60 for the horizontal! Further checking showed this learner did that too.

I wasn’t too sure what to make of it. It’s mathematically correct, so there’s no issue there. The learner has a grasp of the other ratios but is more confident with sine so I can see why he would default to that position, although I hope the extra time it takes isn’t an issue tomorrow. I can’t fathom, however, why the markscheme would show it this way in the first instance. (Not the only time a markscheme has confused me recently!)

*What do you think? Have you got any quirky methods like this? Have any if your students? Do you have an idea why a markscheme would default to this position? I’d love to hear your response.*

I remember being taught this way myself for resolving forces and always thought it was a bit odd. I always tell my students to do their best to avoid using something they have calculated (like a hypotenuse) and work with what is given, but I have seen something similar. I find that students are less happy with using the tan ratio than coz or sin too.

I always stress that to my students too, don’t carry through a potential error if it can be avoided! That’s odd about tan, it was always my go to ratio!

I had a student once who solved *every single quadratic* by completing the square, because she “preferred that one”!

I can sympathise with that. It’s my preferred method of solving a quadratic, although I’ll obviously use factorisation if it’s easy to spot.

I found out about a technique which always uses cos when resolving forces. “Cos to close” when resolving in any two dimensions. Eg, a force of 10N at 37• to the horizontal would be 10cos37 for the horizontal component (closing in on the horizontal axis) and 10cos 53 for the vertical component (closing in on the vertical axes). It works for forces at angles on slopes etc. , ie parallel and perp to the slope. I’ve ended up using it in other ways too.

Aye, cosine is “sine of the complementary angle” so cos (90-x) is interchangeable with cos x (a fact a number of mine forgot when it cane up on Junec2013 c3. I can’t see why this would be more beneficial than using sine though?

Say you’ve got 4 different forces working at a point in different directions, instead of resolving each of them into their horizontal and vertical components and then adding them up to get to the resultant components, which some students get wrong becaus they’ve forgotten which were negative and which were positive, you can “cos” them all to the specified axis and the values will be neg or pos already, then just add. Feel free to try this at home.

I’m still not sure it has any benefit?

I use “cos to close” but only to help them quickly label the adjacent side, then they still use sine for the opposite.

I’m not sure if I’ve ever used or taught tan for resolving components. I try to show mine that they have a perfectly valid choice and to use whichever angle is easier to spot. In a similar vein, I try to give them a choice when it comes to taking moments: sometimes it’s easier to use a perpendicular distance measured directly to the line of action of a force; others it’s easier to use a given distance and then use a perpendicular component of the force. When they’re making the sensible choice everytime you know they’ve really got it.

Aye, certainly agree with moments here. The question he didn’t use tan on was vectors not resolving.

Ah, well that’s just silly 🙂