Question

determine the remaining sides and angles of the triangle abc.
a=112.33°, c=20.46°, c=120

b= 47.21 °

a ≈ 317.6
(do not round until the final answer. then round to the nearest tenth as needed.)
b ≈
(do not round until the final answer. then round to the nearest tenth as needed.)

Explanation:

Step1: Recall the Law of Sines

The Law of Sines states that $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$. We know values of $A$, $B$, $C$, $a$, and $c$, so we can use the ratio $\frac{b}{\sin B}=\frac{c}{\sin C}$ to find $b$.

Step2: Identify known values

We know that $B = 47.21^\circ$, $C = 20.46^\circ$, and $c = 120$.

Step3: Set up the equation for $b$

From the Law of Sines, $b=\frac{c\sin B}{\sin C}$.

Step4: Substitute the values

Substitute $c = 120$, $\sin B=\sin(47.21^\circ)$, and $\sin C=\sin(20.46^\circ)$ into the equation.
First, calculate $\sin(47.21^\circ)\approx0.734$ and $\sin(20.46^\circ)\approx0.349$.
Then, $b=\frac{120\times0.734}{0.349}$.

Step5: Calculate the value

$120\times0.734 = 88.08$. Then, $\frac{88.08}{0.349}\approx252.4$.

Answer:

$b\approx252.4$