document.write( "Question 609058: Find the exact value of the expression 1/2cos(pi/12)+(sqrt3)/2sin(pi/12) \n" ); document.write( "
Algebra.Com's Answer #383514 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
\"%281%2F2%29cos%28pi%2F12%29%2B%28sqrt%283%29%2F2%29sin%28pi%2F12%29\"
\n" ); document.write( "First of all, when you see/hear \"exact value\" in a Trig problem, you should know to put your calculator away. The problem can and must be solved using special angles.

\n" ); document.write( "But \"pi%2F12\" is NOT one of our special angles!? So how are we supposed to figure this out? Well, somehow we have to find a way to change each \"pi%2F12\" into one of the special angles.

\n" ); document.write( "There are a number of Trig properties/identities that allow you to change the argument of a trig function:
  • Angle sum properties: sin(A+B), cos(A+B), tan(A+B)
  • Angle difference properties: sin(A-B), cos(A-B), tan(A-B)
  • The double angle properties: sin(2x), cos(2x) [of which there are 3 varieties), tan(2x)
  • The half angle properties: \"sin%28%281%2F2%29x%29\", \"cos%28%281%2F2%29x%29\" and \"tan%28%281%2F2%29x%29\"
So which one(s) will help us change \"pi%2F12\" into a special angle. There are two clues:
  • None of the properties has a \"sqrt%283%29\" in it. So your expression, as it is written will not fit any of them. We will need to find a way to rewrite your expression so that it fits the pattern in one of the properties.
  • The \"1%2F2\" and the \"sqrt%283%29%2F2\" are both special values:
    \n" ); document.write( "\"1%2F2+=+sin%28pi%2F6%29+=+cos%28pi%2F3%29\" and
    \n" ); document.write( "\"sqrt%283%29%2F2+=+sin%28pi%2F3%29+=+cos%28pi%2F6%29\"
Let's see what we get if we
  • Replace the 1/2 with \"sin%28pi%2F6%29\" and the \"sqrt%283%292\" with \"cos%28pi%2F6%29\":
    \n" ); document.write( "\"sin%28pi%2F6%29cos%28pi%2F12%29+%2B+cos%28pi%2F6%29sin%28pi%2F12%29\"
    \n" ); document.write( "This may look like a step backward but upon closer examination we should see that it matches the pattern of the right side sin(A+B):
    \n" ); document.write( "sin(A+B) = sin(A)cos(B) + cos(A)sin(B)
    \n" ); document.write( "This means we can rewrite
    \n" ); document.write( "\"sin%28pi%2F6%29cos%28pi%2F12%29+%2B+cos%28pi%2F6%29sin%28pi%2F12%29\"
    \n" ); document.write( "as
    \n" ); document.write( "\"sin%28pi%2F6%2Bpi%2F12%29\"
    \n" ); document.write( "This still may not look like progress. But look as what happens when we add the fractions:
    \n" ); document.write( "\"sin%28%282pi%29%2F12%2Bpi%2F12%29\"
    \n" ); document.write( "\"sin%28%283pi%29%2F12%29\"
    \n" ); document.write( "\"sin%28pi%2F4%29\"
    \n" ); document.write( "And presto! we have a special angle! This simplifies to:
    \n" ); document.write( "\"sqrt%282%29%2F2\"
  • Replace the 1/2 with \"cos%28pi%2F3%29\" and the \"sqrt%283%292\" with sin(pi/3):
    \n" ); document.write( "\"cos%28pi%2F3%29cos%28pi%2F12%29+%2B+sin%28pi%2F3%29sin%28pi%2F12%29\"
    \n" ); document.write( "This expression marches the pattern of the right side of cos(A-B):
    \n" ); document.write( "cos(A-B) = cos(A)cos(B) + sin(A)sin(B)
    \n" ); document.write( "so we can replace
    \n" ); document.write( "\"cos%28pi%2F3%29cos%28pi%2F12%29+%2B+sin%28pi%2F3%29sin%28pi%2F12%29\"
    \n" ); document.write( "with
    \n" ); document.write( "\"cos%28pi%2F3+-+pi%2F12%29\"
    \n" ); document.write( "which simplifies as follows:
    \n" ); document.write( "\"cos%28%284pi%29%2F12+-+pi%2F12%29\"
    \n" ); document.write( "\"cos%28%283pi%29%2F12%29\"
    \n" ); document.write( "\"cos%28pi%2F4%29\"
    \n" ); document.write( "\"sqrt%282%29%2F2\"
So either way we get \"sqrt%282%29%2F2\" for an answer.

\n" ); document.write( "NOTE: Trig will be a lot easier if you learn that these properties are patterns. The x's, A's and B's in all these properties are just placeholders. They can be replaced by any mathematical expression and the equation will still be true!! \n" ); document.write( "
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