Question 376518
{{{2*cos^2(75)-1}}}
There are three variations to the formula for cos(2x):<ul><li>{{{cos(2x) = cos^2(x) - sin^2(x)}}}</li><li>{{{cos(2x) = 2*cos^2(x) - 1}}}</li><li>{{{cos(2x) = 1 - 2sin^2(x)}}}</li></ul>
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(3/2)}}} And since 150 is in the second quadrant and cos is negative in the second quadrant...
cos(150) = {{{-sqrt(3)/2}}}<br>
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(3)/2}}}. 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.