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Edexcel A Level Mathematics

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Trigonometric Formulae

In this video, the goal is to solve an equation that uses the compound angle formula. The equation is tan(theta+45) + 2 tan(theta) = 1, and we need to find the solutions in the interval of 0 to 360 degrees. We cannot solve this equation using traditional methods, so we need to use the compound angle formula. The formula is written out, and we substitute in the values for A and B. We can simplify the equation by replacing tan(45) with 1, then multiplying every term by 1-tan(theta), and rearranging the terms. We factorize the equation and find that the solutions are theta=0, 180, 360, and approximately 43.43 or 243.43 degrees. It is important to note that since tan graphs repeat every 180 degrees, we need to add 180 to each solution to find all possible solutions in the given interval.

Hello, hello! So in this video we're trying to solve an equation that uses the compound angle formula. So in this one here it says solve in the interval of 0 to 360 degrees this thing here that tan of theta plus 45 plus 2 tan theta equals 1 and we need to give our answers to two decimal places. So the issue with this is we can't really solve it because here we have a tan of theta and here we have a tan of theta plus 45 so you can't even like factorize anything or there's no way for us to find our solutions. Instead because there's one angle plus another one inside the tan function we're going to have to use a compound angle formula. So the first thing we want to do is write down what that formula is and that's what I've done here. And then we just apply that. Instead of writing this we're going to write all of this stuff. And I suppose it's helpful. Basically we want to think of this here as the A and this here's B. So A is theta, B is 45. So when I actually write all that out, I'm writing this at the beginning, I'm going to have tan of A which in our case is theta plus tan of 45 because that's our B and we're going to divide all of that by 1 minus tan theta tan 45. So we replace this with all of that. Then the 2 tan theta, we can't really do anything with that, that's fine. And that's all going to equal 1. That's great. So now we can simplify it down a bit. So tan 45 is equal to 1. Yeah so you can put this in your calculator or it's one of the ones that you kind of meant to memorize but you don't need to memorize them because you have a calculator. So if we apply that to here, on the top I have tan of theta plus 1 and the bottom we have 1 minus tan theta times 1 so just tan theta. And then the rest of it doesn't really change. We still have plus 2 tan theta and that's still going to be equal to 1. Okay so a bit simpler now and now we've only got tans. Yeah that's pretty good, we just got tan thetas. So to simplify this though, I want to basically get rid of this fraction. Fractions are always a pain. So I'm going to multiply every single term by 1 minus tan theta and that means that over here I will just have tan theta plus 1. Over here, actually I'm just going to make that a bit closer otherwise I'll run out of space. I'm going to have my 2 tan theta multiplied by my 1 minus tan theta. And that's going to equal over the right one set of this so just 1 minus tan theta. Cool. Now I could bring everything to the same side or I can expand this first. We could probably do both in one go really. So we've got tan theta, this one, plus this one, plus and then 2 tan theta times 1 is 2 tan theta. 2 tan theta times minus tan theta is minus 2 tan squared theta. And I'm going to take away the 1 so it's on this side and I'm going to add the tan so it's on this side as well so everything is on the left. Okay so now we've got everything there, now we just need to simplify it all down. So I like to put my squares first so I'm just going to write the minus 2 tan squared theta first. Then I've got 1 tan theta, 2 more tan thetas and another tan theta. So in total I've got 4 tan thetas. And I've got a 1 here and minus 1 here so they cancel out, I actually don't have any integers. That's going to equal 0. So now looking much more simple and now to find out what my solutions are I need to factorize it first. So I'm going to take out a common factor of minus 2 tan theta and in brackets I'll just have a tan theta plus 4. Now it equals 0. No sorry, a mistake there, minus 2. If I've taken out a minus 2 factor on this 4 I only have minus 2 left inside. My bad, sorry. So what we've got from this is tan theta is either 0, that's the solution I get from here. So tan theta is either 0 or from this bit tan theta is 2. So I can put those both into my calculator. This one would say that theta equals, let me just do it. So my calculator needs to be, let's check what we're working with, degrees, degrees here. So for calculating degrees I do tan to the minus 1 of 0, that's just 0. And if we put tan to the minus 1 of 2 that gives me 63.43 degrees. Okay so the last thing we need to do is remember that if it's a tan graph it's the same, it repeats every 180 degrees. So if I'm looking for a range of 0 to 360 then I'm going to have more solutions when I've added 180 to each of these. So 0 add 180 is 180. There would be another solution at 360 but you've got to double check. Oh and this one does go up to 360 and include it actually, so fair. That'd be another one there, 360. I thought it was only less than 360. You've got 3 there. And then for this one as well, I should put a little degree symbol on all these. We can add 180 to this answer and we get 43.43 degrees. And that's it. We have our five solutions. And that's all we've got to do. So I hope that all made sense. If you have any questions about any of that then let us know down in the FAQ. Otherwise have a look at the MCQs at the end of the topic. You can do a lot of practice on this, that's the best way to get better. And I'll see you in the next video.

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