Question

solve each triangle abc that exists.
a = 75.7° a = 12.4 b = 9.2
select the correct choice below and, if necessary, fill in the answer boxes within the choice.
○ a. there is only one possible solution for the triangle.
the measurements for the remaining angles a and c and side c are as follows.
b = □° c = □° c = □
(round to the nearest tenth as needed.)
○ b. there are two possible solutions for the triangle.
the measurements for the solution with the longer side c are as follows.
b₁ = □° c₁ = □° c₁ = □
the measurements for the solution with the shorter side c are as follows.
b₂ = □° c₂ = □° c₂ = □
(round to the nearest tenth as needed.)
○ c. there are no possible solutions for the triangle.

Explanation:

Step1: Use the Law of Sines

By the Law of Sines, $\frac{\sin B}{b}=\frac{\sin A}{a}$. Substitute $A = 75.7^{\circ}$, $a = 12.4$, and $b = 9.2$: $\sin B=\frac{b\sin A}{a}=\frac{9.2\sin75.7^{\circ}}{12.4}$.
Calculate $\sin75.7^{\circ}\approx0.9703$. Then $\sin B=\frac{9.2\times0.9703}{12.4}\approx0.716$.
So $B=\sin^{- 1}(0.716)\approx45.8^{\circ}$ (since $0^{\circ}

Step2: Find angle \(C\)

Since $A + B + C=180^{\circ}$, then $C = 180^{\circ}-A - B$. Substitute $A = 75.7^{\circ}$ and $B = 45.8^{\circ}$: $C=180^{\circ}-75.7^{\circ}-45.8^{\circ}=58.5^{\circ}$.

Step3: Find side \(c\)

Again, by the Law of Sines $\frac{c}{\sin C}=\frac{a}{\sin A}$. So $c=\frac{a\sin C}{\sin A}$. Substitute $a = 12.4$, $A = 75.7^{\circ}$, and $C = 58.5^{\circ}$.
$\sin58.5^{\circ}\approx0.852$, $\sin75.7^{\circ}\approx0.9703$. Then $c=\frac{12.4\times0.852}{0.9703}\approx10.9$.

Answer:

A. There is only one possible solution for the triangle.
The measurements for the remaining angles \(B\) and \(C\) and side \(c\) are as follows.
\(B = 45.8^{\circ}\), \(C = 58.5^{\circ}\), \(c = 10.9\)