Question

solve each triangle abc that exists.
a = 76.6° a = 10.3 b = 9.5

select the correct choice below and, if necessary, fill in the answer boxes within the choice.

a. 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.)

b. 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.)

c. there are no possible solutions for the triangle.

Explanation:

Step1: Apply Law of Sines

Use \(\frac{\sin A}{a}=\frac{\sin B}{b}\) to find \(\sin B\).
\(\sin B=\frac{b\sin A}{a}=\frac{9.5\sin76.6^\circ}{10.3}\)
Calculate \(\sin76.6^\circ\approx0.971\), so \(\sin B\approx\frac{9.5\times0.971}{10.3}\approx\frac{9.2245}{10.3}\approx0.8956\).

Step2: Determine possible \(B\)

Since \(\sin B\approx0.8956\), \(B_1\approx\arcsin(0.8956)\approx63.7^\circ\) (acute). Check if obtuse \(B_2 = 180^\circ - 63.7^\circ = 116.3^\circ\) is valid: \(A + B_2 = 76.6^\circ + 116.3^\circ = 192.9^\circ>180^\circ\), so \(B_2\) is invalid. Thus, only one solution.

Step3: Find angle \(C\)

\(C = 180^\circ - A - B = 180^\circ - 76.6^\circ - 63.7^\circ = 39.7^\circ\).

Step4: Find side \(c\)

Use Law of Sines: \(\frac{c}{\sin C}=\frac{a}{\sin A}\), so \(c=\frac{a\sin C}{\sin A}=\frac{10.3\sin39.7^\circ}{\sin76.6^\circ}\).
\(\sin39.7^\circ\approx0.640\), so \(c\approx\frac{10.3\times0.640}{0.971}\approx\frac{6.592}{0.971}\approx6.8\).

Answer:

B. There is only one possible solution for the triangle.
The measurements for the remaining angles \(B\) and \(C\) and side \(c\) are as follows.
\(B = 63.7^\circ\)
\(C = 39.7^\circ\)
\(c = 6.8\)