SOLUTION: Simplify the expression by using a double-angle formula or a half-angle formula. 7cos^2(theta/9)-7sin^2(theta/9) sin(theta/6)cos(theta/6)

Algebra ->  Trigonometry-basics -> SOLUTION: Simplify the expression by using a double-angle formula or a half-angle formula. 7cos^2(theta/9)-7sin^2(theta/9) sin(theta/6)cos(theta/6)      Log On


   

Question 236921: Simplify the expression by using a double-angle formula or a half-angle formula.
7cos^2(theta/9)-7sin^2(theta/9)
sin(theta/6)cos(theta/6)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
First, we'll need the following formulas for these two problems:
cos%282x%29+=+cos%5E2%28x%29+-+sin%5E2%28x%29
sin%282x%29+=+2sin%28x%29cos%28x%29

To become skilled at using these (and the other Trig. formulas):
  • Look at the x's as only placeholders. The x's can be anything. So, using the first of the two properties above:
    cos%282y%29+=+cos%5E2%28y%29+-+sin%5E2%28y%29
    cos%2820%29+=+cos%5E2%2810%29+-+sin%5E2%2810%29
    cos%282x%2B2y%29+=+cos%5E2%28x%2By%29+-+sin%5E2%28x%2By%29
    cos%28%281%2F5%29x%29+=+cos%5E2%28%281%2F10%29x%29+-+sin%5E2%28%281%2F10%29x%29 (Since 1/5 = 2*(1/10))
  • The coefficients in the formula can often be "matched" by use of factoring. (Both of your problems will need this so after seeing their solutions I hope you understand this tip better.)

Now let's try your problems.
7cos%5E2%28theta%2F9%29-7sin%5E2%28theta%2F9%29
We can use the cos(2x) formula on this because:
  • The formula has cos^2 - sin^2 and so does your expression.
  • The formula has the same arguments (x) for cos^2 and sin^2 and so does your expression (theta/9).
  • The formula has the same coefficients (1) for cos^2 and sin^2 and so does your expression (7).

This is how you figure out if a formula can be used. Before we actually go ahead and use the formula we need to "match" the coefficients. This is done by factoring (sometimes creatively).

Your coefficients are 7's and we want 1's. So factor out a 7:
7%28cos%5E2%28theta%2F9%29+-+sin%5E2%28theta%2F9%29%29
Inside the parentheses we have the exact pattern of the cos(2x) forumula. So we can replace that expression with cos(2*theta/9):
7%28cos%282%2Atheta%2F9%29%29
And we can remove the outer parentheses:
7cos%282%2Atheta%2F9%29

sin%28theta%2F6%29cos%28theta%2F6%29
We can use the sin(2x) formula on this because:
  • The formula has sin * cos and so does your expression.
  • The formula has the same arguments (x) for cos and sin and so does your expression (theta/6).
  • The formula has a coefficient of 2 and your expression can have a coefficient of 2 with some creativity.

Here's how to get the coefficient we need. As you know, multiplying something by one does not change it. To get the 2 we need as a coefficient, we need to create a 1, as a product, which has a two in it. Since the product of reciprocals is always 1, we need (1/2)*2. So we multiply by this giving:
%281%2F2%29%2A2%2Asin%28theta%2F6%29cos%28theta%2F6%29
Using the Associative Property we can group and separate the expression that matches the sin(2x) pattern:
%281%2F2%29%2A%282%2Asin%28theta%2F6%29cos%28theta%2F6%29%29
Now we can replace the pattern with sin(2*theta/6):
%281%2F2%29%2Asin%28theta%2F3%29