SOLUTION: Find the exact value of cos 105º by using a half-angle formula. Thank You

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Question 42455: Find the exact value of cos 105º by using a half-angle formula.
Thank You
Answer by psbhowmick(878) About Me  (Show Source):
You can put this solution on YOUR website!
cos%28105%5Eo%29
= cos%28180%5Eo+-+75%5Eo%29
= -cos%2875%5Eo%29

Now, cos%2875%5Eo%29
= cos%2830%5Eo+%2B+45%5Eo%29
= cos%2830%5Eo%29cos%2845%5Eo%29+-+sin%2830%5Eo%29sin%2845%5Eo%29
= %28sqrt%283%29%2F2%29%2A%281%2Fsqrt%282%29%29+-+%281%2F2%29%2A%281%2Fsqrt%282%29%29
= sqrt%283%29%2F%282%2Asqrt%282%29%29+-+1%2F%282%2Asqrt%282%29%29
= %28sqrt%283%29-1%29%2F%282sqrt%282%29%29

Hence, cos%28105%5Eo%29
= -cos%2875%5Eo%29
= -%28sqrt%283%29-1%29%2F%282sqrt%282%29%29


VVI
No irratonal number can have any exact value. As the answer is an irrational fraction so its value is always approximate. For more detailed discussion on rational and irrational numbers see my lesson on "Rational & Irrational Numbers" at
http://www.algebra.com/tutors/lessons-by-tutor.mpl?tutor=psbhowmick