# trigonometryHow calculate cos of 165 (sum or difference of what angles)? cos(165)

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### 3 Answers

We need to find the values of cos(165).

We will need to rewrite the number (165) as a sum of two known angles.

165 = 210 - 45 (210 is the angle 30 in the third quadrant).

==> cos(165) = cos(210 - 45)

Now we will use trigonometric identities to solve.

cos(210 - 45) = cos210*cos45 + sin210*sin45

= -sqrt3/2 * sqrt2/2 + (-1/2)*(sqrt2/2)

= -sqrt6/ 4 - sqrt2 /4

= (-sqrt6 - sqrt2)/4

= -sqrt2(sqrt3 +1) /4

**==> cos(165) = -sqrt2(sqrt3 +1) / 4**

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 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

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

Please discuss why sum or why other way?