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Small Angle Approximations

This video discusses how to simplify complicated formulas by using small angle approximations for trigonometric functions. The trick is to look for clues in the question indicating when theta is small. The speaker walks through an example problem, demonstrating how to substitute the small angle formulas for the trig functions and simplify the equation. At the end, the equation is rearranged to match the desired form. The video ends with a call to practice and work on MCQs.

Hello, so in this video we're looking at a certain type of question where they give you a kind of complicated looking formula or expression and they ask you to simplify it. And the way that we simplify it is we use our small angle approximation formulas to convert any sine, cos and tan functions just into some form of theta. And the key that is this type of question is that they'll say in the question when theta is small. So that's the clue that you can use your small angle approximations. So in this particular one it says when theta is small show that all of this fairly complicated stuff can be simplified to this here. So the way we're going to do this is we write out our small angle approximation formulas, write out the question and then we're going to plug these things in. So to get started for this one instead of writing cos theta I'm going to write 1 minus theta squared over 2. So 1 minus theta squared over 2. Then I still keep this minus 1. Instead of sine 3 theta I'm just going to put plus 3 theta. Because remember from this if this was a 3 theta here this would be a 3 theta as well. And then the 4 theta squared I don't need to change because that's not a trig function so I can stay as it is. Then down here this 3 theta will stay the same and this tan 2 theta will just become 2 theta. So put it to that. Yeah so we've now substituted all of these into here and simplified it down a bit. And our next step then is to actually simplify. So I have this little approximate symbol again. On the top we have 1, take away 1, these are the only integers we have. 1 minus 1, 0 so they cancel out. Then for the theta squares I have minus theta squared over 2 plus 4 theta squared. So in total that gives me 3 and a half or 3.5 theta squared on top. Yeah 4 theta squared minus theta squared over 2, 3.5 theta squared. And then for my thetas I only have 3 theta. So that's that. Then on the bottom 3 theta minus 2 theta is just a single theta. So that becomes that. And then we can simplify that down a little bit more by dividing everything by theta. So we don't need two lines this time. If I actually divide this and this by theta I'm gonna get 3.5 theta plus 3. Yeah that's pretty good. That to me now seems as simple as it can go. So the last thing we need to do is rearrange it as necessary. So what we actually want to get is one half of 7 theta plus 6. Whereas we have 3.5 theta plus 3. So all they've really done here is take out a factor of one half. So we can do that, that's no problem. We don't really have to do anything different. We just take out a factor of one half and so in brackets we're gonna have 7 theta plus 6, which is exactly what they wanted. And that's it. So remember overall for these you recall all your formulas, you write out the equation, you substitute them into your equation, and then you just simplify it down and finally at the end rearrange it if you need to to get in the particular form that they want. Anyway that's everything for this video. So have a go at the MCQs and just give it a bit of practice and I'll see you in the next video. Thanks!

Using Small Angle Approximations

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