Question

solve the triangle. write each answer as an integer or as a decimal rounded to the nearest tenth. m∠u = ° m∠v = ° w =

Explanation:

Step1: Use the Law of Cosines to find side \(w\)

The Law of Cosines formula is \(w^{2}=u^{2}+v^{2}-2uv\cos W\). Here \(u = 14\), \(v = 8\), and \(W=19^{\circ}\).
\[

$$\begin{align*} w^{2}&=14^{2}+8^{2}-2\times14\times8\times\cos(19^{\circ})\\ w^{2}&=196 + 64-224\times\cos(19^{\circ})\\ w^{2}&=260-224\times0.9455\\ w^{2}&=260 - 211.792\\ w^{2}&=48.208\\ w&\approx6.9 \end{align*}$$

\]

Step2: Use the Law of Sines to find \(\angle U\)

The Law of Sines is \(\frac{\sin U}{u}=\frac{\sin W}{w}\). We know \(u = 14\), \(w\approx6.9\), and \(W = 19^{\circ}\).
\[

$$\begin{align*} \sin U&=\frac{u\sin W}{w}\\ \sin U&=\frac{14\times\sin(19^{\circ})}{6.9}\\ \sin U&=\frac{14\times0.3256}{6.9}\\ \sin U&=\frac{4.5584}{6.9}\\ \sin U&\approx0.6606\\ U&\approx41.4^{\circ}\approx41.4^{\circ} \end{align*}$$

\]

Step3: Find \(\angle V\)

Since the sum of angles in a triangle is \(180^{\circ}\), \(V=180^{\circ}-U - W\).
\[

$$\begin{align*} V&=180^{\circ}- 41.4^{\circ}-19^{\circ}\\ V&=119.6^{\circ}\approx119.6^{\circ} \end{align*}$$

\]

Answer:

\(m\angle U = 41.4^{\circ}\)
\(m\angle V = 119.6^{\circ}\)
\(w = 6.9\)