# What is the sum of functions sin 135+cos150? please explain the sum

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We have to find the sum sin 135 + cos 150

Use the relations: sin (90 + x) = cos x and cos(180 - x) = -cos x

sin 135 + cos 150

=> sin (90 + 45) + cos (180 - 30)

=> cos 45 - cos 30

cos 45 = 1/sqrt 2 and cos 30 = sqrt 3/2

=> 1/sqrt 2 - sqrt 3/2

=> sqrt 2/2 - sqrt 3/2

=> (sqrt 2 - sqrt 3)/2

**The required sum is (sqrt 2 - sqrt 3)/2**

We notice that 135 = 90 + 45 and 150 = 90 + 60

We'll use the trigonometric identities:

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

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

According to all of these, we'll have:

sin 135 = sin (90 + 45) = sin 90*cos 45 + sin 45*cos 90

We'll have sin 90 = 1 ; cos 45 = sin 45 = sqrt2/2 and cos 90 = 0

sin 135 = sqrt2/2

cos 150 = cos (90 + 60) = cos 90*cos60 - sin90*sin60

cos 90 = 0, sin 90 = 1, cos 60 = 1/2, sin 60 = sqrt3/2

cos 150 = -sqrt3/2

**The value of the sum is sin 135+cos 150 = (sqrt2 - sqrt3)/2.**