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Trigonometric identities

Trigonometric identities
10.3
I said earlier that, for some, trigonometry was an unpleasant subject and I would explain why. And I think I know why. It’s due to the fact that, if you look at our functions, sine, cosine, tangent, well, there’s a lot to know about these functions, not just a definition, but there’s specific values and a whole bunch of properties, identities, formulas. And so you need to memorize a whole lot of things. And I think that explains why a lot of people disliked the subject early on. I’m going to summarize here and now the main things that you need to know that everyone should know about these functions if you work with trigonometry. And I’m going to try keep it to a minimum.
54.9
So what do you need to know about these functions? Well, in my opinion, you need to have clearly in your mind an idea of what the graphs look like. The sine function, we’ve seen. The cosine function, between 0 and 2 pi is all you need to know, because they’re periodic of period 2 pi, and the tangent function what its graph looks like between minus pi over 2 and pi over 2. What else do you need to know? Well, it is expected that a certain number of values are such that you need to know these values automatically, like 0, for instance. You should know extremely well, automatically, the sine of 0 is 0.
94.6
The cosine of 0 is 1, and so forth. All of these are, you notice, pretty much pi over 2, or pi, or things that differ from those things by a multiple of pi with the exceptions of the ones involving pi over 6, pi over 4, and pi over 3. But those come from two triangles. So if you remember those two triangles, and if you hearken back to our definition of sine, cosine, and tangent, in terms of triangles– which you should still bear in mind– then you easily have those values from the table available to you. Now how about identities or formulas? Well, here you have to brace yourself.
136.8
I’m going to show you the sort of minimum that everyone should know. Are you ready? Yes, that. The first one, the circular or Pythagorean identity, sine squared plus cos squared equals 1. Then there’s the fact that sine is odd and cosine is even. There are the complementarity formulas that we’ve seen. And then there are the double-angle formulas that I haven’t spoken of earlier. For example, the sine of 2x is twice sine x cos x– beautiful formula– and another formula for the cosine of 2x. There are the sum and difference formulas. For example, the sine of a sum, x plus y, is sine x cos y plus cos x sine y.
182.4
I had a teacher a long time ago who forced us to memorize those. And he was cruel. And he hit us with a ruler on the hand if we didn’t memorize them fast enough. But boy, did he do us a favour. I’ve known these formulas all my life. And it’s been useful. By the way, these sum and difference formulas, they imply all the formulas that you see above them. They even imply a few other things that I haven’t listed in this already long list of identities you should know. For example, they imply the supplementary formula. Two angles are supplementary if they add up to pi, and the sine of two supplementary angles is the same.
223.4
You can deduce that from the sum formula that I mentioned, also a number of other formulas. But this is certainly adequate. If you knew all this, well, I would be impressed. Here’s an example of how you use these kinds of identities. And by the way, in these trigonometric exercises that we’ll see, the key is to have these facts that I’ve listed at hand, to know these formulas, to be able to use them readily. Suppose you want to simplify the expression that you see. How do you do that? Well, one thing you could do is you could change the definition of tangent here. And instead of writing tangent, you could write sine over cosine.
269.8
Putting everything in terms of sine over cosine is often useful. Now I’m going to put things over a common denominator, in this case, cosine x. And you see in the numerator, you get 1 minus sine squared. Immediately, you should be ready to recognize that as being the same as cosine squared. Why? Because of the circular identity. Once you have cos squared over cos, you cancel one out. And you get cosine. And that’s your simplification. Let’s look at another example, typical problem. You want to calculate the sine of pi over 12. Now you want to do this by hand, not with your calculator. Here’s a clever way to do it.
311
You write pi over 12 as pi over 4 minus pi over 6. Why? Because you know by heart the sine of pi over 4, and pi over 6, and also their cosines. You now apply the sine difference formula, this one. You apply it with these data. And you get an expression for the sine of pi over 12, in terms of other sines and cosines. You proceed to use your knowledge of what those sines and cosines are, possibly based on these canonical triangles that everyone remembers. You substitute them in. And you get your answer at the end. Let’s look at a last example. Often, in trigonometry, you want to prove an identity, like this one, for example.
360.2
And one useful technique for proving such a thing is to work on both sides until you get the same thing on either side. So let’s start with the left side. I’m going to replace tangent by sine over cos. I’m going to multiply top and bottom by cos. And I get this. Well, let’s leave that at that for the time being and work on the right-hand side. Again, I replace tangent by sine over cos. I now use the sine sum formula and cosine sum formula. That will bring in sine and cosine of pi over 4. But I know what they are. They are standard values. I put them in. I simplify.
400.6
And I get something which I now recognize as the same expression as what I have on the left-hand side. Therefore, I have proven the identity.

In this video, Francis introduces the identities to remember about sine, cosine, and tangent functions. In particular the Pythagorean identity, and the sum/difference formulas are presented.

You can access a copy of the slides used in the video in the PDF file at the bottom of this step.

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Advanced Precalculus: Geometry, Trigonometry and Exponentials

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