Question

solve the triangle. round all measures to the nearest tenth of a unit. solve the following triangle:

Explanation:

Step1: Use the Law of Cosines to find side $z$.

The Law of Cosines formula is $z^{2}=x^{2}+y^{2}-2xy\cos Z$. Here $x = 4.4$, $y = 5.1$, and $Z=80^{\circ}$. First, calculate $\cos(80^{\circ})\approx0.174$. Then $z^{2}=4.4^{2}+5.1^{2}-2\times4.4\times5.1\times0.174$.
$z^{2}=19.36 + 26.01-2\times4.4\times5.1\times0.174=19.36 + 26.01 - 7.87=37.5$. So $z=\sqrt{37.5}\approx6.1$ m.

Step2: Use the Law of Sines to find angle $Y$.

The Law of Sines is $\frac{\sin Y}{y}=\frac{\sin Z}{z}$. We know $y = 5.1$, $z\approx6.1$, and $\sin Z=\sin(80^{\circ})\approx0.985$. So $\sin Y=\frac{y\sin Z}{z}=\frac{5.1\times0.985}{6.1}=\frac{5.0235}{6.1}\approx0.824$. Then $Y=\sin^{- 1}(0.824)\approx55.5^{\circ}$.

Step3: Find angle $X$.

Since the sum of angles in a triangle is $180^{\circ}$, $X = 180^{\circ}-Y - Z$. Substitute $Y\approx55.5^{\circ}$ and $Z = 80^{\circ}$, so $X=180^{\circ}-55.5^{\circ}-80^{\circ}=44.5^{\circ}$.

Answer:

Side $z\approx6.1$ m, angle $Y\approx55.5^{\circ}$, angle $X\approx44.5^{\circ}$