Problem

In the \(SSA\) triangle shown, \(c~=~12\), \(C~=~70°\), and \(a~=~8\). Use the \(\text{Law}~\text{of}~\text{Sines}\) to find every valid value of \(A\), check each resulting angle sum, and state the number of possible triangles.

An SSA ray, fixed base, swing circle, altitude, and unlabeled geometric intersections for evaluating inverse-sine candidates.

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Reverse the ratio to get sin(A)=12sin(70 degrees)/8, about 1.410, and report 0 triangles.

Keep only A=141.2 degrees; then B=180-141.2-70=-31.2 degrees, but still report 1 triangle.

sin(A)=8sin(70 degrees)/12 gives A=38.8 degrees or 141.2 degrees; accept both without the angle-sum test and report 2 triangles.

sin(A)=8sin(70 degrees)/12, so A=38.8 degrees or 141.2 degrees. Since 141.2+70 exceeds 180, only A=38.8 degrees remains; B=71.2 degrees and 1 triangle is possible.

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In the \(SSA\) triangle shown, \(c~=~12\), \(C~=~70°\), and \(a~=~8\). Use the \(\text{Law}~\text{of}~\text{Sines}\) to find every valid value of \(A\), check each resulting angle sum, and state the number of possible triangles.

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