Solving Equations Involving Cosine Plus Sine

Here's a trick for solving equations that involve cos+sin. The trick is to square both sides, because that way, some identities make it turn out a lot easier to solve. But you have to be careful when doing this because you might introduce extra solutions. Here's an example. I find it nicest to first divide both sides by something so that the left side is cosx+sinx. Then the trick is to square both sides and then expand this out. If you do that there is a sin² +cos² that becomes 1, and for the middle term you can use the sin double angle formula. (See  Double Angles and Half Angles.) Then when you subtract 1 from both sides you get a simple equation involving sin2x. From this the idea is to solve for 2x and then divide by 2 to get x. But you have to be a bit careful with this, because if the problem asks you to find all solutions in [0,2pi), that means that x is in [0,2pi), not 2x. For x to be in [0,2pi) it isn't necessary for 2x to be there, 2x can be a bit bigger and still have x in [0,2pi), namely it only needs to be in [0,4pi), so to find all the possible solutions we have to go around the unit circle twice. Doing this we get Then dividing by 2 we get But since we squared both sides we have to check each of these, because we could have introduced extra solutions. With multiples of pi/12 this could be tricky, though, because they are not special angles. You could use the half angle formulas, but that is still a bit of a pain. You might find it simplest just to use your calculator, and it is fine with me if you do it that way, but do realize, of course, that there may be round off error. The nicer way and the way I would do it is simply to estimate by using the unit circle. The idea behind this is that really the only kind of extra solutions you will introduce by squaring both sides are ones where cosx+sinx is off by a minus sign from what it is supposed to be, so all you really have to check in this problem is that cosx+sinx is positive. 7pi/12 is 1 pi/12 more than 6pi/12=pi/2, so it is in the second quadrant and closer to the pi/2 than to pi, so cosine is little negative, but sine is big positive, (there is more blue than red) so cos+sin is positive, so this one is okay. 11pi/12 is just under pi, so it is in the second quadrant and closer to pi than to pi/2, so cosine is big negative and sine is little positive, (there is more red than blue) so cos+sin is negative, so this one we throw out. 19pi/12 is just 1 pi/12 more than 18pi/12=3pi/2, so it is in the 4th quadrant and closer to 3pi/2 than to 3pi, so cosine is little positive and sine is bit negative, (there is more red than blue) so cos+sin is negative, so this one goes out too. 23pi/12 is just 1 pi/12 less than 2pi, so it is in the 4th quadrant and closer to 2pi than to 3pi/2, so sine is little negative and cosine is big positive, (there is more blue than red) so cos+sin is positive, so this one we keep. So we get a final solution of
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