Question

In Exercises 17–32, two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively.
a = 95, c = 125, A = 49°

Accepted Answer

Identify the given elements: side $$a = 95$$, side $$c = 125$$, and angle $$A = 49^\circ. $$Note that angle $$A is $$opposite side $$a. $$Use the Law of Sines to find angle $$C or to $$check the number of possible triangles. The Law of Sines states: $$\frac{a}{\sin A} = \frac{c}{\sin C}. $$Rearrange the Law of Sines to solve for $$\sin C$$: $$\sin C = \frac{c \cdot \sin A}{a}. $$Calculate $$\sin C$$ using the given values (do not find the final value yet). Then analyze the value of $$\sin C to $$determine the number of possible triangles: if $$\sin C \u003e 1$$, no triangle; if $$\sin C = 1$$, one right triangle; if $$0 \u003c \sin C \u003c 1$$, two possible triangles (since $$\sin is $$positive in two quadrants). If one or two triangles exist, find angle $$C by $$taking $$\arcsin(\sin C)$$ for the first solution, and for the second solution use $$180^\circ - C. $$Then find angle $$B$$ using the triangle angle sum $$B = 180^\circ - A - C$$, and finally find side $$b$$ using the Law of Sines: $$\frac{b}{\sin B} = \frac{a}{\sin A}.$$

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Law of Sines

Ambiguous Case of SSA Triangles

Triangle Solution and Rounding