Question

what is the value of z, rounded to the nearest tenth? use the law of sines to find the answer. law of sines: $\frac{sin(a)}{a}=\frac{sin(b)}{b}=\frac{sin(c)}{c}$

Explanation:

Step1: Find the third - angle.

The sum of angles in a triangle is $180^{\circ}$. Let the third angle be $\angle A$. So, $\angle A=180-(75 + 51)=54^{\circ}$.

Step2: Apply the law of sines.

We know that $\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$. Using $\frac{\sin(75^{\circ})}{z}=\frac{\sin(54^{\circ})}{2.6}$.
We know that $\sin(75^{\circ})=\sin(45^{\circ}+ 30^{\circ})=\sin45^{\circ}\cos30^{\circ}+\cos45^{\circ}\sin30^{\circ}=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\times\frac{1}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\approx0.966$ and $\sin(54^{\circ})\approx0.809$.
From $\frac{\sin(75^{\circ})}{z}=\frac{\sin(54^{\circ})}{2.6}$, we can cross - multiply to get $z\times\sin(54^{\circ})=2.6\times\sin(75^{\circ})$.
Then $z=\frac{2.6\times\sin(75^{\circ})}{\sin(54^{\circ})}$.

Step3: Calculate the value of z.

Substitute the values of $\sin(75^{\circ})$ and $\sin(54^{\circ})$: $z=\frac{2.6\times0.966}{0.809}=\frac{2.5116}{0.809}\approx3.1$. Rounding to the nearest tenth, $z\approx3.2$.

Answer:

3.2 units