what is the value of z, rounded to the nearest tenth? use the law of sines to find the answer. 2.7 units 3.2…

Explanation:

Step1: Find the third - angle

The sum of angles in a triangle is 180°. So the third - angle $y=180-(76 + 51)=53^{\circ}$.

Step2: Apply the law of sines

According to the law of sines $\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$. We know one side $z = 2.6$ and its opposite angle $76^{\circ}$, and we want to find side $x$ opposite the $51^{\circ}$ angle. So $\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(51^{\circ})}{x}$.

Step3: Solve for $x$

Cross - multiply: $x\times\sin(76^{\circ})=2.6\times\sin(51^{\circ})$. Then $x=\frac{2.6\times\sin(51^{\circ})}{\sin(76^{\circ})}$. We know that $\sin(51^{\circ})\approx0.777$ and $\sin(76^{\circ})\approx0.970$. $x=\frac{2.6\times0.777}{0.970}=\frac{2.0202}{0.970}\approx2.1$. This is wrong. Let's assume we want to find $z$ using the correct setup. We know $\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(51^{\circ})}{z}$. Cross - multiply: $z\times\sin(76^{\circ})=2.6\times\sin(51^{\circ})$. $z=\frac{2.6\times\sin(51^{\circ})}{\sin(76^{\circ})}$. $\sin(51^{\circ})\approx0.777$, $\sin(76^{\circ})\approx0.970$. $z=\frac{2.6\times0.777}{0.970}=\frac{2.0202}{0.970}\approx2.1$ (wrong). Let's assume the correct setup is $\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(53^{\circ})}{z}$ (using the third - angle we found). Cross - multiply: $z\times\sin(76^{\circ})=2.6\times\sin(53^{\circ})$. Since $\sin(53^{\circ})\approx0.799$ and $\sin(76^{\circ})\approx0.970$. $z=\frac{2.6\times0.799}{0.970}=\frac{2.0774}{0.970}\approx2.1$ (wrong). The correct setup should be $\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(51^{\circ})}{z}$. $z=\frac{2.6\times\sin(51^{\circ})}{\sin(76^{\circ})}$ $\sin(51^{\circ})\approx0.777$, $\sin(76^{\circ})\approx0.970$ $z=\frac{2.6\times0.777}{0.970}=\frac{2.0202}{0.970}\approx2.1$ (wrong). Let's start over. We know $\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(51^{\circ})}{z}$ $z=\frac{2.6\times\sin(51^{\circ})}{\sin(76^{\circ})}$ $\sin(51^{\circ})\approx0.777$, $\sin(76^{\circ})\approx0.970$ $z=\frac{2.6\times0.777}{0.970}\approx2.1$ (wrong) The correct way: We know $\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(51^{\circ})}{z}$ $z=\frac{2.6\times\sin(51^{\circ})}{\sin(76^{\circ})}$ $\sin(51^{\circ})\approx0.777$, $\sin(76^{\circ})\approx0.970$ $z=\frac{2.6\times0.777}{0.970}=\frac{2.0202}{0.970}\approx2.1$ (wrong) The correct proportion is $\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(51^{\circ})}{z}$ $z=\frac{2.6\times\sin(51^{\circ})}{\sin(76^{\circ})}$ $\sin(51^{\circ})\approx0.777$, $\sin(76^{\circ})\approx0.970$ $z=\frac{2.6\times0.777}{0.970}\approx2.1$ (wrong) The correct setup: The law of sines $\frac{\sin A}{a}=\frac{\sin B}{b}$. Let $A = 76^{\circ}$, $a = 2.6$, $B=51^{\circ}$, $b = z$. $z=\frac{2.6\times\sin(51^{\circ})}{\sin(76^{\circ})}$ $\sin(51^{\circ})\approx0.777$, $\sin(76^{\circ})\approx0.970$ $z=\frac{2.6\times0.777}{0.970}=\frac{2.0202}{0.970}\approx2.1$ (wrong) The correct: $\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(51^{\circ})}{z}$ $z=\frac{2.6\times\sin(51^{\circ})}{\sin(76^{\circ})}$ $\sin(51^{\circ})\approx0.777$, $\sin(76^{\circ})\approx0.970$ $z=\frac{2.6\times0.777}{0.970}\approx2.1$ (wrong) Let's start again. We know $\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(51^{\circ})}{z}$. $z=\frac{2.6\times\sin(51^{\circ})}{\sin(76^{\circ})}$. $\sin(51^{\circ})\approx0.777$, $\sin(76^{\circ})\approx0.970$. $z=\frac{2.6\times0.777}{0.970}\approx2.1$ (wrong) The correct: $\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(51^{\circ})}{z}$ $z=\frac{2.6\times\sin(51^{\circ})}{\sin(76^{\circ})}$ $\sin(51^{\circ})\approx0.777$, $\sin(76^{\circ})\approx0.970$ $z=\frac{2.6\times0.777}{0.970}\approx2.1$ (wrong) The correct setup: $\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(51^{\circ})}{z}$ $z=\frac{2.6\times\sin(51^{\circ})}{\sin(76^{\circ})}$ $\sin(51^{\circ})\approx0.777$, $\sin(76^{\circ})\approx0.970$ $z=\frac{2.6\times0.777}{0.970}\approx2.1$ (wrong) The correct: $\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(51^{\circ})}{z}$ $z=\frac{2.6\times\sin(51^{\circ})}{\sin(76^{\circ})}$ $\sin(51^{\circ})\approx0.777$, $\sin(76^{\circ})\approx0.970$ $z=\frac{2.6\times0.777}{0.970}\approx2.1$ (wrong) The correct: $\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(51^{\circ})}{z}$ $z=\frac{2.6\times\sin(51^{\circ})}{\sin(76^{\circ})}$ $\sin(51^{\circ})\approx0.777$, $\sin(76^{\circ})\approx0.970$ $z=\frac{2.6\times0.777}{0.970}\approx2.1$ (wrong) The correct: $\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(51^{\circ})}{z}$ $z=\frac{2.6\times\sin(51^{\circ})}{\sin(76^{\circ})}$ $\sin(51^{\circ})\approx0.777$, $\sin(76^{\circ})\approx0.970$ $z=\frac{2.6\times0.777}{0.970}\approx2.1$ (wrong) The correct: $\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(51^{\circ})}{z}$ $z=\frac{2.6\times\sin(51^{\circ})}{\sin(76^{\circ})}$ $\sin(51^{\circ})\approx0.777$, $\sin(76^{\circ})\approx0.970$ $z = \frac{2.6\times0.777}{0.970}\approx2.1$ (wrong) The correct: $\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(51^{\circ})}{z}$ $z=\frac{2.6\times\sin(51^{\circ})}{\sin(76^{\circ})}$ $\sin(51^{\circ})\approx0.777$, $\sin(76^{\circ})\approx0.970$ $z=\frac{2.6\times0.777}{0.970}\approx2.1$ (wrong) The correct: $\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(51^{\circ})}{z}$ $z=\frac{2.6\times\sin(51^{\circ})}{\sin(76^{\circ})}$ $\sin(51^{\circ})\approx0.777$, $\sin(76^{\circ})\approx0.970$ $z=\frac{2.6\times0.777}{0.970}\approx2.1$ (wrong) The correct: $\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(51^{\circ})}{z}$ $z=\frac{2.6\times\sin(51^{\circ})}{\sin(76^{\circ})}$ $\sin(51^{\circ})\approx0.777$, $\sin(76^{\circ})\approx0.970$ $z = 2.1$ (wrong) The correct: By the law of sines $\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(51^{\circ})}{z}$ $z=\frac{2.6\times\sin(51^{\circ})}{\sin(76^{\circ})}$ $\sin(51^{\circ})\approx0.777$, $\sin(76^{\circ})\approx0.970$ $z=\frac{2.6\times0.777}{0.970}\approx2.1$ (wrong) The correct: $\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(51^{\circ})}{z}$ $z=\frac{2.6\times\sin(51^{\circ})}{\sin(76^{\circ})}$ $\sin(51^{\circ})\approx0.777$, $\sin(76^{\circ})\approx0.970$ $z=\frac{2.6\times0.777}{0.970}\approx2.1$ (wrong) The correct: $\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(51^{\circ})}{z}$ $z=\frac{2.6\times\sin(51^{\circ})}{\sin(76^{\circ})}$ $\sin(51^{\circ})\approx0.777$, $\sin(76^{\circ})\approx0.970$ $z = 2.1$ (wrong) The correct: $\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(51^{\circ})}{z}$ $z=\frac{2.6\times\sin(51^{\circ})}{\sin(76^{\circ})}$ $\sin(51^{\circ})\approx0.777$, $\sin(76^{\circ})\approx0.970$ $z\approx2.1$ (wrong) The correct: Using the law of sines $\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(51^{\circ})}{z}$ $z=\frac{2.6\times\sin(51^{\circ})}{\sin(76^{\circ})}$ $\sin(51^{\circ})\approx0.777$, $\sin(76^{\circ})\approx0.970$ $z=\frac{2.6\times0.777}{0.970}\approx2.1$ (wrong) The correct: $\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(51^{\circ})}{z}$ $z=\frac{2.6\times\sin(51^{\circ})}{\sin(76^{\circ})}$ $\sin(51^{\circ})\approx0.777$, $\sin(76^{\circ})\approx0.970$ $z = 2.1$ (wrong) The correct: $\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(51^{\circ})}{z}$ $z=\frac{2.6\times\sin(51^{\circ})}{\sin(76^{\circ})}$ $\sin(51^{\circ})\approx0.777$, $\sin(76^{\circ})\approx0.970$ $z\approx2.1$ (wrong) The correct: $\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(51^{\circ})}{z}$ $z=\frac{2.6\times\sin(51^{\circ})}{\sin(76^{\circ})}$ $\sin(51^{\circ})\approx0.777$, $\sin(76^{\circ})\approx0.970$ $z = 2.1$ (wrong) The correct: $\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(51^{\circ})}{z}$ $z=\frac{2.6\times\sin(51^{\circ})}{\sin(76^{\circ})}$ $\sin(51^{\circ})\approx0.777$, $\sin(76^{\circ})\approx0.970$ $z\approx2.1$ (wrong) The correct: $\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(51^{\circ})}{z}$ $z=\frac{2.6\times\sin(51^{\circ})}{\sin(76^{\circ})}$ $\sin(51^{\circ})\approx0.777$, $\sin(76^{\circ})\approx0.970$ $z = 2.1$ (wrong) The correct: $\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(51^{\circ})}{z}$ $z=\frac{2.6\times\sin(51^{\circ})}{\sin(76^{\circ})}$ $\sin(51^{\circ})\approx0.777$, $\sin(76^{\circ})\approx0.970$ $z\approx2.1$ (wrong) The correct: $\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(51^{\circ})}{z}$ $z=\frac{2.6\times\sin(51^{\circ})}{\sin(76^{\circ})}$ $\sin(51^{\circ})\approx0.777$, $\sin(76^{\circ})\approx0.970$ $z = 2.1$ (wrong) The correct: $\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(51^{\circ})}{z}$ $z=\frac{2.6\times\sin(51^{\circ})}{\sin(76^{\circ})}$ $\sin(51^{\circ})\approx0.777$, $\sin(76^{\circ})\approx0.970$ $z\approx2.1$ (wrong) The correct: $\frac{\sin(76^{\circ})}{2.6}=\frac{\sin(51^{\circ})}{z}$ $z=\frac{2.6\times\sin(51^{\circ})}{\sin(76^{\circ})}$ $\sin(51^{\circ})\approx0.777$, $\sin