Question

Find the unknown angles in triangle ABC for each triangle that exists.B = 74.3°, a = 859 m, b = 783 m

Accepted Answer

Identify the given elements: angle $$B = 74.3^\circ$$, side $$a = 859 m ($$opposite angle $$A$$), and side $$b = 783 m ($$opposite angle $$B). We $$need to find the unknown angles $$A$$ and $$C in $$triangle $$ABC. $$Use the Law of Sines, which states: $$\frac{a}{\sin A} = \frac{b}{\sin B}. $$Substitute the known values to set up the equation: $$\frac{859}{\sin A} = \frac{783}{\sin 74.3^\circ}. $$Solve for $$\sin A by $$rearranging the equation: $$\sin A = \frac{859 	imes \sin 74.3^\circ}{783}. $$Calculate the right-hand side to find the value of $$\sin A (do $$not compute the final value here). Determine the possible values of angle $$A by $$taking the inverse sine (arcsin) of $$\sin A. $$Remember that the sine function is positive in the first and second quadrants, so there may be two possible angles for $$A$$: one acute and one obtuse (if valid). For each possible value of $$A$$, find angle $$C$$ using the triangle angle sum property: $$C = 180^\circ - A - B. $$Check if the resulting triangle is valid (all angles positive and sum to 180°). This will give you all possible triangles that satisfy the given conditions.

Find the unknown angles in triangle ABC for each triangle that exists.

B = 74.3°, a = 859 m, b = 783 m

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Key Concepts

Law of Sines

Triangle Angle Sum Property

Ambiguous Case of the Law of Sines (SSA Condition)