Find the exact value of sin ⁡ ( x + y ) \sin\left(x+y\right) sin ( x + y ) . Use a trigonometric identity. cos ⁡ x = 5 11 \cos x=\frac{5}{11} cos x = 11 5 , sin ⁡ y = − 1 8 \sin y=-\frac18 sin y = − 8 1 , x x x in fourth quadrant, y y y in third quadrant

− 5 + 12 42 88 \frac{-5+12\sqrt{42}}{88} 88 − 5 + 12 42 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

12 42 71 \frac{12\sqrt{42}}{71} 71 12 42 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

− 12 42 71 -\frac{12\sqrt{42}}{71} − 71 12 42 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

5 − 12 42 88 \frac{5-12\sqrt{42}}{88} 88 5 − 12 42 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z">

Sum and Difference Identities

6. Trigonometric Identities and More Equations / Sum and Difference Identities / Problem 4

Find the exact value of $\sin\left(x+y\right)$ . Use a trigonometric identity.
$\cos x=\frac{5}{11}$ , $\sin y=-\frac18$ , $x$ in fourth quadrant, $y$ in third quadrant

$\frac{12\sqrt{42}}{71}$

$\frac{-5+12\sqrt{42}}{88}$

$-\frac{12\sqrt{42}}{71}$

$\frac{5-12\sqrt{42}}{88}$

Learn this concept

Find the exact value of sin⁡(x+y)\sin\left(x+y\right)sin(x+y). Use a trigonometric identity.cos⁡x=511\cos x=\frac{5}{11}cosx=115​, sin⁡y=−18\sin y=-\frac18siny=−81​, xxx in fourth quadrant, yyy in third quadrant

Find the exact value of $\sin\left(x+y\right)$ . Use a trigonometric identity.
$\cos x=\frac{5}{11}$ , $\sin y=-\frac18$ , $x$ in fourth quadrant, $y$ in third quadrant