Question

In Exercises 1–12, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. If no triangle exists, state 'no triangle.' If two triangles exist, solve each triangle. B = 66°, a = 17, c = 12

Accepted Answer

Identify the given elements of the triangle: angle $$B = 66^\circ$$, side $$a = 17$$, and side $$c = 12. We $$need to find the remaining sides and angles of the triangle. Use the Law of Sines, which states $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$, to find angle $$C or $$angle $$A. $$Since we know $$a$$, $$c$$, and $$B$$, start by finding angle $$C$$ using $$\frac{a}{\sin A} = \frac{c}{\sin C} or $$find $$A$$ first by relating $$a$$ and $$B. $$Calculate angle $$A$$ using the Law of Sines: $$\frac{a}{\sin A} = \frac{c}{\sin C}$$ can be rearranged to find $$\sin A if $$you find $$\sin C$$ first, or alternatively use $$\frac{a}{\sin A} = \frac{b}{\sin B} if$$ $$b is $$known. Since $$b is $$unknown, try to find $$\sin A by $$using $$\frac{a}{\sin A} = \frac{c}{\sin C}$$ and express $$\sin C in $$terms of $$B$$ and $$A. $$Check for the possibility of two triangles by considering the ambiguous case of the Law of Sines (SSA configuration). Calculate $$\sin A$$ and verify if it is less than or equal to 1. If $$\sin A \u003c 1$$, then two possible angles for $$A$$ exist: $$A$$ and $$180^\circ - A. $$This means two triangles may be possible. Once angles $$A$$ and $$C$$ are found, use the fact that the sum of angles in a triangle is $$180^\circ to $$find the missing angle. Then use the Law of Sines again to find the missing side $$b. $$Round all lengths to the nearest tenth and angles to the nearest degree as required.

In Exercises 1–12, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. If no triangle exists, state 'no triangle.' If two triangles exist, solve each triangle. B = 66°, a = 17, c = 12

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Concetti chiave

Law of Sines

Ambiguous Case of the Law of Sines (SSA Condition)

Triangle Angle Sum Property