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The Half-Angle Identities - Problem 1
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
Let's do a problem that uses the half-angle formula. Evaluate sine of pi over 8 using a half-angle formula. Well let's recall the sine half-angle formula. It's sine of theta over 2 equals plus or minus the square root of 1 minus cosine theta all over 2.
Now don't forget we have this plus or minus here that means in the end we have to determine whether our answer is positive or negative, but for now sine of pi over 8, theta in this case would be pi over 4, so we have plus or minus square root of 1 minus cosine pi over 4 all over 2.
Now cosine of pi over 4 is root 2 over 2, so we have plus or minus 1 minus root 2 over 2 all divided by 2. Now this is not a great way to leave your answer because it's not only a complex fraction, but it's got a radical in the denominator. So we want to rationalize the denominator. We can actually do that by working inside the radical, let me show you.
Let's multiply by 2 over 2 inside the radical that's fair because we're multiplying by 1 so we have plus or minus the square root of and this 2 distributes over the top, you get 2 times 1 which is 2 and 2 times root 2 over 2 which is root 21 all over 4 and that gives me plus or minus the square root of 2 minus root 2 over 2.
Now which is it? Positive or negative? Let's draw out our unit circle, pi over 8 is definitely in the first quadrant and so both sine and cosine are going to be positive, so in this case we definitely need a positive answer plus root 2 minus root 2 all over 2.
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