Question

solve for the remaining sides and angles if possible (if not, answer \dne\ in all answer boxes). as in the text, $(a,a),(b,b)$, and $(c,c)$ are angle - side opposite pairs. round to two decimal places, if necessary.
$a = 30.3^{circ},b = 86.6^{circ},a = 5$
$a=30.30^{circ}$
$b = 86.60^{circ}$
$c = 63.10^{circ}$
$a = 5$
$b =$
$c =$

Explanation:

Step1: Find angle \(C\)

Sum of angles in a triangle is \(180^{\circ}\). So \(C = 180^{\circ}-A - B\).
\(C=180^{\circ}-30.3^{\circ}-86.6^{\circ}=63.1^{\circ}\)

Step2: Find side \(b\) using the Law of Sines \(\frac{a}{\sin A}=\frac{b}{\sin B}\)

\(b=\frac{a\sin B}{\sin A}\)
Substitute \(a = 5\), \(A = 30.3^{\circ}\), \(B = 86.6^{\circ}\)
\(\sin A=\sin(30.3^{\circ})\approx0.504\), \(\sin B=\sin(86.6^{\circ})\approx0.998\)
\(b=\frac{5\times0.998}{0.504}\approx9.92\)

Step3: Find side \(c\) using the Law of Sines \(\frac{a}{\sin A}=\frac{c}{\sin C}\)

\(c=\frac{a\sin C}{\sin A}\)
Substitute \(a = 5\), \(A = 30.3^{\circ}\), \(C = 63.1^{\circ}\)
\(\sin C=\sin(63.1^{\circ})\approx0.892\)
\(c=\frac{5\times0.892}{0.504}\approx8.85\)

Answer:

\(b = 9.92\), \(c = 8.85\)