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Answer 1 · 2018-02-28T08:14:51

#sin(a+b)=21/221#

The key is to apply the sum/difference identities

#1. sin(a+b)=sin(a)cos(b)+sin(b)cos(a)#

#2. cos(a-b)=cos(a)cos(b)+sin(a)sin(b)#

#3. tan(a-b)=(tan(a)-tan(b))/(1+tan(a)tan(b))#

We know #cos(a)=12/13# and #sin(b)=8/17#

By the pythagorean trigonometric identity
notice how the sign behaves depending on the quadrant

#sin(a)=sqrt(1-(12/13)^2)=5/13#

#cos(b)=-sqrt(1-(8/17)^2)=-15/17#

First exercise

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

#=5/13*(-15/17)+8/17*12/13#

#=-75/221+96/221#

#=21/221#

The two remaining are left for you, as it follow the same method