Question

consider a triangle \\(abc\\) like the one below. suppose that \\(c = 39\\), \\(b = 34\\), and \\(c = 43^\circ\\). (the figure is not drawn to scale.) solve the triangle.

carry your intermediate computations to at least four decimal places, and round your answers to the nearest tenth.

if no such triangle exists, enter
o solution.\ if there is more than one solution, use the button labeled \or\.

\\(b = \box^\circ\\), \\(a = \box^\circ\\), \\(a = \box\\)

Explanation:

Identify given values and target unknowns

Using the Solving Triangles knowledge point
Given:

• \(c = 39\)
• \(b = 34\)
• \(C = 43^\circ\)

Target:

• Find angle \(B\), angle \(A\), and side \(a\).

Apply the Law of Sines to find angle B

Using the Law of Sines and Ambiguous Case of Law of Sines knowledge points

$$ \frac{\sin B}{b} = \frac{\sin C}{c} \implies \sin B = \frac{b \sin C}{c} $$

$$ \sin B = \frac{34 \sin(43^\circ)}{39} \approx \frac{34 \times 0.681998}{39} \approx 0.594511 $$

Since \(\sin B \approx 0.594511 < 1\), solutions exist.

$$ B_1 = \arcsin(0.594511) \approx 36.4748^\circ \approx 36.5^\circ $$

Check the obtuse alternative:

$$ B_2 = 180^\circ - 36.4748^\circ = 143.5252^\circ $$

Since \(B_2 + C = 143.5252^\circ + 43^\circ = 186.5252^\circ > 180^\circ\), \(B_2\) is invalid.
Thus, there is exactly one unique solution:

$$ B \approx 36.5^\circ $$

Calculate angle A

Using the Solving Triangles knowledge point

$$ A = 180^\circ - B - C $$

$$ A = 180^\circ - 36.4748^\circ - 43^\circ = 100.5252^\circ \approx 100.5^\circ $$

Calculate side a

Using the Law of Sines knowledge point

$$ \frac{a}{\sin A} = \frac{c}{\sin C} \implies a = \frac{c \sin A}{\sin C} $$

$$ a = \frac{39 \sin(100.5252^\circ)}{\sin(43^\circ)} \approx \frac{39 \times 0.983154}{0.681998} \approx 56.2214 \approx 56.2 $$

Answer:

Consider a triangle \(ABC\) like the one below. Suppose that \(c = 39\), \(b = 34\), and \(C = 43^\circ\). Solve the triangle.

\(B =\) <blank>\(36.5\)</blank>\(^\circ\), \(A =\) <blank>\(100.5\)</blank>\(^\circ\), \(a =\) <blank>\(56.2\)</blank>