Question

Solve each triangle ABC that exists.A = 42.5°, a = 15.6 ft, b = 8.14 ft

Accepted Answer

Identify the given elements of the triangle: angle $$A = 42.5^\circ$$, side $$a = 15.6 ft ($$opposite angle $$A$$), and side $$b = 8.14 ft ($$opposite angle $$B). We $$need to find the remaining parts of the triangle: angle $$B$$, angle $$C$$, and side $$c. $$Use the Law of Sines to find angle $$B. $$The Law of Sines states: $$\frac{a}{\sin A} = \frac{b}{\sin B}. $$Rearranged to solve for $$\sin B$$, it becomes $$\sin B = \frac{b \cdot \sin A}{a}. $$Calculate $$\sin B$$ using the values of $$a$$, $$b$$, and $$A. $$Then, find angle $$B by $$taking the inverse sine (arcsin) of $$\sin B. $$Remember that the sine function can have two possible angles in the range $$0^\circ to$$ $$180^\circ$$, so consider both possible solutions for $$B ($$the ambiguous case). Once angle $$B is $$found, calculate angle $$C$$ using the fact that the sum of angles in a triangle is $$180^\circ$$: $$C = 180^\circ - A - B. $$Finally, use the Law of Sines again to find side $$c$$: $$\frac{c}{\sin C} = \frac{a}{\sin A}$$, which rearranges to $$c = \frac{a \cdot \sin C}{\sin A}. $$This completes the solution of the triangle.

Solve each triangle ABC that exists.
A = 42.5°, a = 15.6 ft, b = 8.14 ft

Verified video answer for a similar problem:

Key Concepts

Law of Sines

Triangle Ambiguity (SSA Case)

Sum of Angles in a Triangle