Question

Find the value ? `sin105^@ + cos105^@`

Answer 1

`1/sqrt2`

Explanation:

`sin105^@ + cos105^@`

`sin(60^@+45^@) + cos(60^@+45^@)`

Sum identities:
`sin(alpha +beta) = sin(alpha)cos(beta) +cos(alpha)sin(beta)`
`cos(alpha +beta) = cos(alpha)cos(beta) -sin(alpha)sin(beta)`

`sin60^@ = sqrt3/2`

`sin45^@=sqrt2/2`

`cos60^@=1/2`

`cos45^@=sqrt2/2`

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

Answer 2

`sqrt2/2`

Explanation:

Use trig identity:
`sin x + cos x = sqrt2cos (x - 45)`
In this case:
`sin 105 + cos 105 = sqrt2cos (105 - 45) = sqrt2cos (60)`
Trig table gives `cos 60 = 1/2`, therefor:
`sin 105 + cos 105 = sqrt2/2`