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

2 Answers
May 20, 2018

1/sqrt212

Explanation:

sin105^@ + cos105^@sin105+cos105

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

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

sin60^@ = sqrt3/2sin60=32

sin45^@=sqrt2/2sin45=22

cos60^@=1/2cos60=12

cos45^@=sqrt2/2cos45=22

(sqrt3/2*sqrt2/2+1/2*sqrt2/2) + (1/2*sqrt2/2 -sqrt3/2*sqrt2/2 ) = 1/sqrt2(3222+1222)+(12223222)=12

May 21, 2018

sqrt2/222

Explanation:

Use trig identity:
sin x + cos x = sqrt2cos (x - 45)sinx+cosx=2cos(x45)
In this case:
sin 105 + cos 105 = sqrt2cos (105 - 45) = sqrt2cos (60)sin105+cos105=2cos(10545)=2cos(60)
Trig table gives cos 60 = 1/2cos60=12, therefor:
sin 105 + cos 105 = sqrt2/2sin105+cos105=22