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. C = 50°, a = 3, c = 1

Accepted Answer

Identify the given elements of the triangle: angle $$C = 50^\circ$$, side $$a = 3$$, and side $$c = 1. We $$need to solve the triangle, which means finding the remaining angles $$A$$ and $$B$$, and side $$b. $$Use the Law of Sines, which states: $$\frac{a}{\sin A} = \frac{c}{\sin C}. $$Substitute the known values to set up the equation: $$\frac{3}{\sin A} = \frac{1}{\sin 50^\circ}. $$Solve for $$\sin A by $$cross-multiplying: $$\sin A = 3 	imes \frac{\sin 50^\circ}{1} = 3 \sin 50^\circ. $$Calculate $$3 \sin 50^\circ to $$check if it is less than or equal to 1, which determines if a triangle exists. If $$\sin A is $$greater than 1, then no triangle exists. If $$\sin A is $$less than or equal to 1, find angle $$A by $$taking the inverse sine: $$A = \sin$$^{-1}$$(3 \sin 50^\circ). $$Remember that the sine function can have two possible angles in the range $$0^\circ to$$ $$180^\circ$$, so consider both possible values for $$A to $$check if two triangles exist. Once you find the possible values for $$A$$, calculate angle $$B$$ using the triangle angle sum property: $$B = 180^\circ - A - C. $$Then use the Law of Sines again to find side $$b$$: $$\frac{b}{\sin B} = \frac{a}{\sin A}. $$Round all lengths to the nearest tenth and angles to the nearest degree.

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. C = 50°, a = 3, c = 1

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Key Concepts

Law of Sines

Ambiguous Case of SSA Triangles

Triangle Angle Sum Property