PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
We're talking about the cosine of a difference formula. If you recall it's cosine of beta minus alpha equals cosine beta, cosine alpha plus sine beta, sine alpha. And the way most people will remember this is cosine, cosine, sine, sine and the signs change with the cosine formulas.
Now how can we use that in this problem? It says simplify the expression cosine of 80 degrees times cosine of 20 degrees plus sine of 80 times sine 20. This is going to require the cosine of a difference in reverse. I basically have an expression like this where my beta is 80 degrees and my alpha is 20.
So this is exactly this expression and this would become cosine of 80 degrees minus 20 degrees or 60, the cosine of 60 degrees. Now because cosine of 60 degrees is a special value, I should evaluate that, cosine of 60 is cosine of pi over 3 which is a half so this whole thing simplifies to one-half.
Now in the second example, find the exact value of cosine of 15 degrees, again I can use the cosine of the difference formula. Here I want to think about two numbers whose difference is 15, two special angles, so let me use 60 degrees minus 45 degrees. I know the sine and cosine of 60 and I know the sine and cosine of 40, so this is going to be cosine 60, cosine 45 cosine, cosine-cosine and then sine 60, sine 45 sine, sine-sine and the minus sine becomes a plus sign with a cosine formulas. Think 'c' for 'change sign'.
The cosine of 60 degrees one-half, the cosine of 45 root 2 over 2 plus sine of 60 root 3 over 2 and sine of 45 also root 2 over 2. This gives me root 2 over 4 plus root 6 over 4, which you can also write as root 2 plus root 6 all over 4.
If the problem calls for an exact value, this is the kind of value that your teacher wants. It's exact in the sense that this is not an approximation, this symbol represents the exact value of the cosine of 15 degrees, so you'll see lots and lots of problems like these after you learn the cosine addition formulas.