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 = 1.4, b = 2.9, A = 142°

Accepted Answer

Identify the given elements: side $$a = 1.4$$, side $$b = 2.9$$, and angle $$A = 142^\circ. $$Note that angle $$A is $$opposite side $$a. $$Check the type of triangle and the possibility of solutions using the Law of Sines. The Law of Sines states: $$\frac{a}{\sin A} = \frac{b}{\sin B}. $$Calculate $$\sin B$$ using the formula $$\sin B = \frac{b \sin A}{a}. $$Substitute the known values: $$\sin B = \frac{2.9 	imes \sin 142^\circ}{1.4}. $$Analyze the value of $$\sin B to $$determine the number of possible triangles: if $$\sin B \u003e 1$$, no triangle exists; if $$\sin B = 1$$, one right triangle exists; if $$0 \u003c \sin B \u003c 1$$, two possible triangles may exist (ambiguous case). If one or two triangles exist, find angle $$B by $$taking $$\sin$$^{-1}$$(\sin B)$$, then find angle $$C$$ using $$C = 180^\circ - A - B. $$Finally, use the Law of Sines again to find side $$c$$ with $$c = \frac{a \sin C}{\sin A}.$$

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Law of Sines

Ambiguous Case of SSA Triangles

Angle and Side Measurement Constraints