Math Made Easy: Problem of the Day 17
We haven’t really played with trigonometric identities much yet, so let’s do that today. The key to solving most problems with trigonometric functions in them, is understanding your trigonometric identities, as well as the relationships between the various functions. Something that may look difficult may prove not to be so if you can find ways to manipulate the expression to look more like something you’re familiar with, especially an identity.
Consider today’s problem. Try to evaluate this expression with no calculator. Remember you can always use the unit circle – if you don’t have one, a google search will find you plenty of them.
We see that π/8 is not on our unit circle. So how can we evaluate this without a calculator? Let’s see if we can produce any identities. The first thing I’m going to do, is I’m going to rewrite tangent and cotangent in terms of sine and cosine. After doing that, I want to make this one fraction, so I’m going to give them a common denominator by multiplying the first one by sine over sine and the second by cosine over cosine, giving me the expression at the end here. Note that I’m not writing (θ) for a lot of my work as, when an angle is not specified in trigonometric functions, it’s assumed to be the same angle for all functions involved. I’ll put it back in later when it’s necessary.
I think I’m starting to see something more familiar. Look at the top, it ALMOST looks like a double angle for cosine, but not quite. What if I get sneaky and multiply the whole thing by 1? Then the signs reverse, and I actually DO have a double angle for cosine up top. Now let’s look at the bottom. It is almost the double angle for sine, but not quite. I’m going to be even sneakier now. By multiplying the bottom by ½ AND 2 at the same time, I’m not changing it at all because they cancel, BUT I can now SEE a double angle for sine. And so I replace it with such. And now that I’ve replaced top and bottom with double angles, I pull that ½ out of the bottom, it becomes a big 2, and I can rewrite the whole thing as just 2cot(θ).
So now that I have that, I can plug my π/8 back in. Multiplied by 2, that’s just π/4, and that IS on the unit circle. The cot is the cosine over the sine, as we already showed, and those are both √2/2 at π/4 radians. That evaluates to 1, so our original expression evaluates to 2. We’re done. No calculator needed whatsoever.
Today’s problem was all about remembering your trig idents, and being clever with little ways to multiply things by one that doesn’t actually change the value of your expression, but changes its appearance so you can better recognize what it is, and more easily manipulate it.
