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The Double-Angle Formulas - Problem 2
Here is a slightly more challenging example that involves the sine and cosine in double angle formulas. Problem says if sine of alpha equals 3/5 and alpha is between pi over 2 and pi find cosine of alpha sine of 2 alpha and cosine of 2 alpha.
First thing I want to do is locate alpha on the units circle. Now alpha is between pi over 2 and pi that puts us in the second quadrant and they need to find out where the sine of alpha would be 3/5 and remember sine of alpha gives us the second coordinate on the unit circle so 3/5 it's about here and so going across, that point will be about here. And over the x coordinate yet but the y coordinate is 3/5.
So this is my alpha. The first thing I want to do is find the value of x because that’s going to be the cosine of alpha and I need that, so remember we are on the unit circle so x² plus y² equals 1. Now when we substitute in this point here I get x² plus 3/5 squared equals 1 that’s x² plus 9 over 25 equals 25 over 25. I subtract x² equals 16 over 25 and I get x equals plus or minus 4/5. Well is it plus or minus? In the second quadrant, the x coordinates are going to have to be negative, so let’s take -4/5.
So now we know the cosine and the sine of alpha, that’s one of our answers, cosine alpha is -4/5. Let's find the sine of 2 alpha first. Remember the double angle formula for sine it's 2 sine alpha cosine alpha. Sine of alpha was 3/5 that was given, and the cosine of alpha -4/5. And so we het 25 in the denominator in the numerator I have -12 times 2 negative 12 times 4. So the sine of 2 alpha -24 over 25. And now the cosine 2 alpha.
Cosine of 2 alpha is cosine squared minus sine squared the cosine of alpha was -4/5 so we have to square that and the sine of alpha was 3/5 we have to square that too, so that gives me 16 over 25 minus 9 over 25 and that’s 7 over 25. That’s cosine 2 alpha. So cosine alpha -4/5 sine of 2 alpha -24 over 25 and cosine 2 alpha 7/5.
What’s interesting about this is even if you’ve not drawn a really great diagram of your angle you can see that because cosine is positive and sine is -2 alpha is going to be somewhere over here.