# Evaluate cos 165.

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We that cos (180 - A) = -cos A.

cos ( 165) = cos( 180 - 15) = -cos 15.

cos 2A = (cos A)^2 - (sin A)^2

= 2*(cos A)^2 - 1

cos 30 = 2* (cos 15)^2 - 1

sqrt 3/2 = 2*(cos 15)^2 -1

=> 2*(cos 15)^2 = 1 + sqrt 3/2

=> (cos 15)^2 = 1/2 + sqrt 3/4

=> (cos 15)^2 = (2 + sqrt 3)/4

=> cos 15 = sqrt (2 + sqrt 3)/ 2

=> cos 165 = -sqrt (2 + sqrt 3)/ 2

**So cos 165 = -sqrt (2 + sqrt 3)/ 2**

cos165 degree = - cos (180-165) = -cos15...(1).

Let cos15 = x

Then cos30 = cos2*15 = cos^2(15) -1 = cos 30 = (3^1/2))/2

2x^2-1= (3^1/2)/2

x^2 = (sqrt3)/2 +1

x= sqrt(sqrt3)+2)/2 = 0.9659

Therefore cos165 = -cos15 = - 0.9659 nearly.

We'll write 165 as sum of 2 angles: 90 and 75.

cos(165) = cos (90 + 75)

We'll use the formula:

cos (a+b) = cos a*cos b - sin a*sin b

Comparing, we'll get:

cos (90 + 75) = cos 90*cos 75 - sin 90*sin 75

But cos 90 = 0 and sin 90 = 1

cos (90 + 75) = -sin 75

We'll write 75 as a sum of 2 known angles: 30+45

sin 75 = sin(30+45)

sin(30+45) = sin30*cos45 + sin45*cos30

sin(30+45) = (sqrt2)*(1+sqrt3)/4

cos(165) = -sin 75

**The value of cos(165) = -(sqrt2)*(1+sqrt3)/4**