Question

two sides and an angle are given determine whether the given information results in one triangle, two triangles, or no triangle at all solve any resulting triangle(s)
a = 4, b = 3, a = 40°
select the correct choice and, if necessary, fill in the answer boxes to complete your choice
oa. a single triangle is produced, where b ≈ □°, c ≈ □°, and c ≈ □
(type integers or decimals rounded to two decimal places as needed)

Explanation:

Step1: Use the Law of Sines

The Law of Sines is \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). Given \(a = 4\), \(b = 3\), \(A=40^{\circ}\), we first find \(B\) using \(\frac{a}{\sin A}=\frac{b}{\sin B}\). So \(\sin B=\frac{b\sin A}{a}\).
Substitute the values: \(\sin B=\frac{3\sin40^{\circ}}{4}\).
Since \(\sin40^{\circ}\approx0.6428\), then \(\sin B=\frac{3\times0.6428}{4}=\frac{1.9284}{4}=0.4821\).
So \(B=\sin^{- 1}(0.4821)\approx28.82^{\circ}\) (since \(B\) is acute, \(0^{\circ} b\)).

Step2: Find angle \(C\)

We know that \(A + B+C=180^{\circ}\). So \(C = 180^{\circ}-A - B\).
Substitute \(A = 40^{\circ}\) and \(B\approx28.82^{\circ}\), then \(C=180^{\circ}-40^{\circ}-28.82^{\circ}=111.18^{\circ}\).

Step3: Find side \(c\)

Using the Law of Sines \(\frac{a}{\sin A}=\frac{c}{\sin C}\). So \(c=\frac{a\sin C}{\sin A}\).
Since \(\sin C=\sin111.18^{\circ}\approx0.9336\) and \(\sin A=\sin40^{\circ}\approx0.6428\), \(a = 4\).
Then \(c=\frac{4\times0.9336}{0.6428}=\frac{3.7344}{0.6428}\approx5.81\).

Answer:

A single triangle is produced, where \(B\approx28.82^{\circ}\), \(C\approx111.18^{\circ}\), and \(c\approx5.81\)