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The Double-Angle Formulas - Problem 2

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

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.

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