Question

solve each triangle abc that exists.
a = 77.8° a = 10.9 b = 9.8
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 Law of Sines for angle B

$\sin B = \frac{b \sin A}{a} = \frac{9.8 \sin 77.8^\circ}{10.9}$
Calculate $\sin 77.8^\circ \approx 0.9781$, so $\sin B \approx \frac{9.8 \times 0.9781}{10.9} \approx 0.887$
$B \approx \arcsin(0.887) \approx 62.6^\circ$
Check supplement: $180^\circ - 62.6^\circ = 117.4^\circ$. $A + 117.4^\circ = 77.8^\circ + 117.4^\circ = 195.2^\circ > 180^\circ$, so only one B.

Step2: Calculate angle C

$C = 180^\circ - A - B = 180^\circ - 77.8^\circ - 62.6^\circ = 39.6^\circ$

Step3: Use Law of Sines for side c

$c = \frac{a \sin C}{\sin A} = \frac{10.9 \sin 39.6^\circ}{\sin 77.8^\circ}$
$\sin 39.6^\circ \approx 0.637$, so $c \approx \frac{10.9 \times 0.637}{0.9781} \approx 7.1$

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 = 62.6°, C = 39.6°, c = 7.1 (Round to the nearest tenth as needed.)