Question

solve each triangle abc that exists.
b = 135.5°, c = 8.959, b = 15.841

select the correct choice below and, if necessary, fill in the answer boxes with in the choice.
○ a. there is only one possible solution for the triangle.
the measurements for the remaining angles a and c and side a are as follows.
a = □° c = □° a = □
(round to the nearest (round to the nearest (round to the nearest
tenth as needed.) tenth as needed.) thousandth as needed.)
○ b. there are two possible solutions for the triangle.
the measurements for the solution with the longer side a are as follows.
a₁ = □° c₁ = □° a₁ = □
(round to the nearest (round to the nearest (round to the nearest
tenth as needed.) tenth as needed.) thousandth as needed.)
the measurements for the solution with the shorter side a are as follows.
a₂ = □° c₂ = □° a₂ = □
(round to the nearest (round to the nearest (round to the nearest
tenth as needed.) tenth as needed.) thousandth as needed.)
○ c. there are no possible solutions for the triangle.

Explanation:

Step1: Use the Law of Sines to find angle \(C\)

The Law of Sines is given by \(\frac{\sin B}{b}=\frac{\sin C}{c}\).
We know that \(B = 135.5^{\circ}\), \(b = 15.841\), and \(c=8.959\).
Substitute the values into the formula: \(\sin C=\frac{c\sin B}{b}\)
\(\sin C=\frac{8.959\times\sin(135.5^{\circ})}{15.841}\)
\(\sin(135.5^{\circ})=\sin(180 - 44.5^{\circ})=\sin(44.5^{\circ})\approx0.699\)
\(\sin C=\frac{8.959\times0.699}{15.841}\approx\frac{6.262}{15.841}\approx0.396\)
\(C=\sin^{- 1}(0.396)\approx23.3^{\circ}\)

Step2: Find angle \(A\)

Since the sum of angles in a triangle is \(A + B + C=180^{\circ}\)
\(A = 180^{\circ}-B - C\)
\(A=180^{\circ}-135.5^{\circ}-23.3^{\circ}=21.2^{\circ}\)

Step3: Use the Law of Sines to find side \(a\)

Again, using the Law of Sines \(\frac{\sin A}{a}=\frac{\sin B}{b}\)
\(a=\frac{b\sin A}{\sin B}\)
\(\sin A=\sin(21.2^{\circ})\approx0.361\), \(\sin B=\sin(135.5^{\circ})\approx0.699\), \(b = 15.841\)
\(a=\frac{15.841\times0.361}{0.699}\approx\frac{5.719}{0.699}\approx8.182\)

Answer:

A. There is only one possible solution for the triangle.
The measurements for the remaining angles \(A\) and \(C\) and side \(a\) are as follows.
\(A = 21.2^{\circ}\) (Round to the nearest tenth as needed.)
\(C=23.3^{\circ}\) (Round to the nearest tenth as needed.)
\(a = 8.182\) (Round to the nearest thousandth as needed.)