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

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### 2 Answers

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]**

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.**