Question

How can the value of cos 15 be found without the use of a calculator. Which formula has to be used?

Accepted Answer

The value of cos 30 is usually known to all students.
cos 30 = (sqrt 3)/2.
Now we use the formula for cos 2x, which is:
cos 2x = (cos x)^2 - (sin x)^2
=> cos 2x =  2 (cos x)^2 - 1
Now substitute 15 for x here.
We have cos (30) = 2 (cos 15)^2 - 1
=>  (sqrt 3)/2 = 2 (cos 15)^2 - 1
=> 1 + (sqrt 3)/2 = 2 (cos 15)^2
=> (1 + (sqrt 3)/2)/2 = (cos 15)^2
=> cos 15 = sqrt [ (1 + (sqrt 3)/2)/2]
=> cos 15 = sqrt [ 1/2 + (sqrt 3)/4]
You will most probably need a calculator to find the value of the square root here, but you won't need one which has values of cosine stored in it.
Therefore cos 15 can be calculated without a calculator as sqrt [ 1/2 + (sqrt 3)/4]

Answer

There are so many ways.
We know that cos(A-B) = cosA*cosB + sin A*sinB.
Therefore  cos15 = cos(45-30) = cos45*cos30 + sin45*sin30.
cos 15 =  cos45*cos30 + sin45*sin15 = {1/2^(1/2)}* 1/2+ 1/2^(1/2))*(3^(1/2))/2
cos 15 = 1/2^(1/2){1/2 + 3^(1/2)} = {2^(1/2) + 6^(1/2)}/4 = 0.9659 nearly.
Also we can use cos2*15 = cos 30. Or 2(cos15)^2 -1 = 1/2, Or the solution of 2x^2 -1 = (3^(1/2)}/2,  or x^2 = {2+3^(1/2)}/4.
So x = cos 15 = {[2+3^(1/2)]^(1/2)}/2 = 0.9659 nearly.
So cos 15 = {2^(1/2) + 6^(1/2)}/4 = 0.9659 nearly or cos 15 = {[2+3^(1/2)]^(1/2)}/2 = 0.9659 nearly.

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