SOLUTION: find the exact value using the sum, difference, half, or double angle formulas and the exact value char in cos 105 degrees.

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Question 34344: find the exact value using the sum, difference, half, or double angle formulas and the exact value char in cos 105 degrees.
Answer by longjonsilver(2297) About Me  (Show Source):
You can put this solution on YOUR website!
cos(105) is the same as -cos(75)
--> cos75 = cos(60+15)

Now, cos(60+15) = cos60cos15 - sin60sin15. The trouble is getting the 15 degrees.

+sinA+=+sqrt%28%281%2F2%29+-+%281%2F2%29cos%282A%29%29+
+cosA+=+sqrt%28%281%2F2%29+%2B+%281%2F2%29cos%282A%29%29+

--> +sin15+=+sqrt%28%281%2F2%29+-+%281%2F2%29cos%2830%29%29+
--> +cos15+=+sqrt%28%281%2F2%29+%2B+%281%2F2%29cos%2830%29%29+
--> +sin15+=+sqrt%28%281%2F2%29+-+%281%2F2%29%28sqrt%283%29%2F2%29%29+
--> +cos15+=+sqrt%28%281%2F2%29+%2B+%281%2F2%29%28sqrt%283%29%2F2%29%29+

Now you have all the info you need to make a real mess :-)

Put this all together and you should get that cos75 = +%28sqrt%282%2Bsqrt%283%29%29+-+sqrt%283%29sqrt%282-sqrt%283%29%29+%29%2F4+. If you work this out on your calculator, it is 0.258819.

The answer you require is the negative version of this, since cos115 is in the second quadrant and COS is negative here.

Enjoy!

jon.