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Edexcel A Level Mathematics

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Small Angle Approximations

Small angle approximations allow us to estimate trigonometric functions like sine, cosine, and tangent, without a calculator. The angles involved must be small and in radians, usually at around 0.6 or less. There are three formulas to know: sine theta is approximately equal to theta, cos theta is approximately equal to 1 minus theta squared over 2, and tan theta is equal to theta. The idea is to use smaller values of theta for more accuracy, as larger values produce less reliable approximations. The reason why these approximations work is that for small angles, the graphs of y equals sine theta, y equals cosine theta and y equals tangent theta are almost the same as y equals theta.

Hello and welcome to our topic introduction for small angle approximations. So the aim of this video is to introduce the idea of what's going on in this topic and give you some of the theory or understanding behind it. So to start off, small angle approximations basically allow us to estimate trigonometric functions. So they allow us to estimate things like sine theta or cos theta, even if we don't have a calculator and we can't use the proper sine or cosine functions. But in order to do it there's two really important criteria. The first is that the angles involved must be small, so the theta, when you do sine theta, has to be a fairly small value. It wouldn't work if theta was like 2 or 3. Generally it only works if theta is about 0.6 or less. There isn't any kind of hard cutoff but generally the bigger theta is, the less reliable these approximations are. And the other point is that the angle has to be in radians. So it doesn't work if you're using degrees. If you're using degrees you're going to want to convert that into radians first and then use the formulas that we'll look at. So there's three formulas to know. The first one is that sine theta is approximately equal to theta. So whatever this value is, as long as it's small, sine of that angle will just be the angle itself. And we'll see an example in a minute. Then the next one, a bit harder this time, is that cos theta is approximately equal to 1 minus theta squared over 2. And tan theta is equal to theta. So if we see how these things actually work. The example is if you wanted to find sine of 0.3 but you didn't have a calculator so you couldn't actually put it into the calculator, it's roughly going to be about 0.3. And you can try this now, you can pause the video, put it into your calculator. It wouldn't be exactly 0.3 but it would be very very close. Which is why we use this kind of wavy equal sign instead. It just means it's approximately equal to, or roughly the same as. If you want to find cosine of 0.3 though, it'd be a bit different. This time we'd have to do 1, like it says here, minus 0.3 squared over 2. And that would give us 0.955. So it's a bit harder but you can still do this without a calculator if you really needed to. And you can certainly do this with a simple calculator that didn't have these trig functions on. And finally tan, an easy one again, tan of 0.3 would just be 0.3 again. You can try that in the calculator and see if it works. The idea though is that as these values get bigger, like if you tried this for sine of 0.4, 0.5, 0.6, it would get less and less accurate as your value of theta gets bigger. That's something very important to bear in mind. And so the last thing I want to show you in this video is just why these work. So if we just get up a graph, this is the Desmos website, and I'm going to show you here the graph for y equals sine theta. Yeah, so hopefully you recognize this. We can see it's going kind of down and up, just like a normal sine graph. The x-axis here is your theta or your x, it doesn't matter. I put thetas in here because we're using radians but they could be x's as well. And if I show you now y equals theta, so this is a straight line through the origin, you can see that for these values here, when x or theta is quite small, they're actually the same. So the green line and the blue line are basically touching each other, they're running alongside each other. And so if I want to know what the y value is when my theta value is like 0.3, it's just going to be 0.3 because I can just use this y equals theta equation instead. Yeah, this equation is just as basically going to give me the answers just as well as this one is for these small values. Same for small negative values to be fair. If we look at the cosine ones, here's a cosine graph, and if I show you a 1 minus theta squared over 2 graph, it also looks almost exactly the same for this region here. Whenever x is quite small, they look basically the same. So instead of using a complicated cosine function, we can get away with just using this function, and we'll get a pretty good approximation as long as we're using smaller values of theta. And finally the tan one, this is the tan graph, but if I show you the y equals theta graph, again pretty close for these small values of x. And that's why we can use the approximations. So I hope this all made sense. You actually don't need to fully understand all this graph stuff in order to do the questions at all, but I think it helps and it's better to understand stuff if you can. If it still doesn't make sense, let us know down in the FAQ and we'll explain it a bit more. But otherwise I'll see you in the other videos. Thanks!

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