Question

Solve each triangle. See Examples 2 and 3.B = 74.8°, a = 8.92 in., c = 6.43 in.

Accepted Answer

Identify the given elements of the triangle: angle $$B = 74.8^\circ$$, side $$a = 8.92$$ inches (opposite angle $$A$$), and side $$c = 6.43$$ inches (opposite angle $$C). We $$need to find the remaining angles $$A$$ and $$C$$, and side $$b. $$Use the Law of Sines, which states: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}. $$Since we know $$a$$, $$c$$, and $$B$$, we can find angle $$C$$ first by setting up the ratio $$\frac{a}{\sin A} = \frac{c}{\sin C} or $$use $$\frac{c}{\sin C} = \frac{a}{\sin A}$$, but since $$A is $$unknown, it's easier to use $$\frac{c}{\sin C} = \frac{a}{\sin A}$$ after finding $$C$$ from $$B$$ and sides. To find angle $$C$$, use the Law of Sines ratio between sides $$a$$ and $$c$$ and their opposite angles $$A$$ and $$C. $$Since $$A is $$unknown, instead use the Law of Cosines or find angle $$C by $$first finding angle $$A$$ using the Law of Sines with known $$a$$, $$B$$, and $$c. $$Alternatively, use the Law of Sines with $$a$$, $$B$$, and $$c to $$find $$C by $$setting up $$\frac{a}{\sin A} = \frac{c}{\sin C}$$ and expressing $$A in $$terms of $$B$$ and $$C. $$Once you find angle $$C$$, calculate angle $$A$$ using the fact that the sum of angles in a triangle is $$180^\circ$$: $$A = 180^\circ - B - C. $$Finally, use the Law of Sines again to find side $$b by $$setting $$\frac{b}{\sin B} = \frac{a}{\sin A}$$ and solving for $$b.$$

Solve each triangle. See Examples 2 and 3.

B = 74.8°, a = 8.92 in., c = 6.43 in.

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Concetti chiave

Law of Sines

Triangle Angle Sum Property

Ambiguous Case of the Law of Sines (SSA)