SOLUTION: 2 cos squared (75degrees)-1

Algebra ->  Trigonometry-basics -> SOLUTION: 2 cos squared (75degrees)-1       Log On


   

Question 376518: 2 cos squared (75degrees)-1
Answer by jsmallt9(3759) About Me  (Show Source):
You can put this solution on YOUR website!
2%2Acos%5E2%2875%29-1
There are three variations to the formula for cos(2x):
  • cos%282x%29+=+cos%5E2%28x%29+-+sin%5E2%28x%29
  • cos%282x%29+=+2%2Acos%5E2%28x%29+-+1
  • cos%282x%29+=+1+-+2sin%5E2%28x%29

Your expression fits the middle variation with x = 75. Using the pattern for this formula, with 75 in the place of x, we get:
cos(2*75)
which simplifies to
cos(150)
The reference angle for 150, since it is in the second quadrant, is 180 - 150 or 30. This is one of the special angles whose sin's and cos's you are supposed to learn. cos(30) = sqrt%283%2F2%29 And since 150 is in the second quadrant and cos is negative in the second quadrant...
cos(150) = -sqrt%283%29%2F2

Note: If you simply use your calculator on the original expression, you will end up with a decimal. It would be difficult to recognize this decimal as sqrt%283%29%2F2. The decimal is an approximate answer. The square root answer is 100% exactly correct. The moral of the story, if you can use special angles on a problem, do so. You will end up with exact answers instead of decimal approximations. This is why we used the cos(2x) formula. It turned and expression with a non-special angle, 75, into an expression involving a special angle, 150.