SOLUTION: Hi, this of one of two problems I just can't seem to work out. The directions say "Verify the Identity". {{{ sin(4x) = 4sin(x)cos^3(x) - 4cos(x)sin^3(x) }}} Thanks so much!!

Algebra ->  Trigonometry-basics -> SOLUTION: Hi, this of one of two problems I just can't seem to work out. The directions say "Verify the Identity". {{{ sin(4x) = 4sin(x)cos^3(x) - 4cos(x)sin^3(x) }}} Thanks so much!!      Log On


   

Question 817638: Hi, this of one of two problems I just can't seem to work out. The directions say "Verify the Identity".
+sin%284x%29+=+4sin%28x%29cos%5E3%28x%29+-+4cos%28x%29sin%5E3%28x%29+
Thanks so much!!
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
When verifying identities some of the things to do are:
  • Match the arguments of the trig functions.
  • Mathch the names of the trig functions.
  • Match the number of terms.
With +sin%284x%29+=+4sin%28x%29cos%5E3%28x%29+-+4cos%28x%29sin%5E3%28x%29+ we can see that the argument on the left is 4x but on the right the arguments are x's. So we know that we have to change the argument from 4x to x.

Changing arguments of trig functions are done through trig identities. In Trig you learn a lot of identities:
  • sin%5E2%28x%29%2Bcos%5E2%28x%29+=+1
  • cos%28A%2BB%29+=+cos%28A%29cos%28B%29-sin%28A%29sin%28B%29
  • sin%282x%29+=+2sin%28x%29cos%28x%29
  • etc.
Half the battle is memorizing them. The other half is learning how to use them effectively. The key to learning how to use them is understanding that the x's, A's and B's of these identities may be replaced by any valid mathematical expression!.

So even though we have no well-known identity for sin(4x), we do have an identity we can use on it. The identity sin(2x) = 2sin(x)cos(x) can be used. It tells us that the sine of anything is equal to 2 times (the sine of half of the anything) times the cosine of half of the anything. So we can use it on sin(4x) (where "anything" = 4x and "half of anything" = 2x:
+2sin%282x%29cos%282x%29+=+4sin%28x%29cos%5E3%28x%29+-+4cos%28x%29sin%5E3%28x%29+
We still have some argument changing to do. The 2x's need to be x's. We can use sin(2x) again (more obviously this time) to change the argument of sin(2x). We can use cos(2x) to change the argument of cos(2x) to x. There are three variations of cos(2x):
  • cos%282x%29+=+cos%5E2%28x%29-sin%5E2%28x%29
  • cos%282x%29+=+2cos%5E2%28x%29-1
  • cos%282x%29+=+1-2sin%5E2%28x%29
Any of them may be used here. By looking ahead I can see that the first one will make things a little easier. So, using sin(2x) and cos(2x) on the left side we get:

The arguments are now x's. Let's multiply this out to see how close we are to the end:


And look at how close we got! All we have to do is use the Commutative Property on the second term to get the factors in the same order as they are on the right: