`sin(u + v)` Find the exact value of the trigonometric expression given that sin(u) = -7/25 and cos(v) = -4/5 (Both u and v are in quadrant III.)

Given `sin(u)=-7/25 , cos(v)=-4/5`

using pythagorean identity,

`sin^2(u)+cos^2(u)=1`

plug in the value of sin(u),

`(-7/25)^2+cos^2(u)=1`

`49/625+cos^2(u)=1`

`cos^2(u)=1-49/625`

`cos^2(u)=(625-49)/625`

`cos^2(u)=576/625`

`cos(u)=sqrt(576/625)`

`cos(u)=+-24/25`

Since u is in Quadrant III ,

`:.cos(u)=-24/25`

Now `sin^2(v)+cos^2(v)=1`

plug in the value of cos(v)=-4/5,

`sin^2(v)+(-4/5)^2=1`  

`sin^2(v)+16/25=1`

`sin^2(v)=1-16/25=9/25`

`sin(v)=sqrt(9/25)`

`sin(v)=+-3/5`

since v is in Quadrant III ,

`:.sin(v)=-3/5`

`sin(u+v)=sin(u)cos(v)+cos(u)sin(v)`

plug in the values ,

`sin(u+v)=(-7/25)(-4/5)+(-24/25)(-3/5)`

`sin(u+v)=28/125+72/125=100/125`

`sin(u+v)=4/5`

Given `sin(u)=-7/25 , cos(v)=-4/5`

using pythagorean identity,

`sin^2(u)+cos^2(u)=1`

plug in the value of sin(u),

`(-7/25)^2+cos^2(u)=1`

`49/625+cos^2(u)=1`

`cos^2(u)=1-49/625`

`cos^2(u)=(625-49)/625`

`cos^2(u)=576/625`

`cos(u)=sqrt(576/625)`

`cos(u)=+-24/25`

Since u is in Quadrant III ,

`:.cos(u)=-24/25`

Now `sin^2(v)+cos^2(v)=1`

plug in the value of cos(v)=-4/5,

`sin^2(v)+(-4/5)^2=1`  

`sin^2(v)+16/25=1`

`sin^2(v)=1-16/25=9/25`

`sin(v)=sqrt(9/25)`

`sin(v)=+-3/5`

since v is in Quadrant III ,

`:.sin(v)=-3/5`

`sin(u+v)=sin(u)cos(v)+cos(u)sin(v)`

plug in the values ,

`sin(u+v)=(-7/25)(-4/5)+(-24/25)(-3/5)`

`sin(u+v)=28/125+72/125=100/125`

`sin(u+v)=4/5`