Problem

For the SSA triangle shown, \(A~=~30°\), \(a~=~10~m\), and \(b~=~15~m\). Find every valid triangle, including \(B\), \(C\), \(c\), and the total number of solutions.

Prompt acute SSA construction with A=30°, opposite side a=10 m, side b=15 m, altitude, and swing circle; the triangle count and solutions are withheld.

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An SSA solution is not complete after the calculator returns one inverse-sine angle. Test that angle and its supplement separately, rejecting any candidate that leaves a nonpositive third angle. For every surviving configuration, keep its remaining angle and side together as one matched set, and report every complete set rather than mixing measurements across cases.

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Reverse the ratio to get sin(B)=10sin(30°)/15≈0.333; use B≈19.5° and 160.5°, including C≈-10.5°, and report 2 triangles.

sin(B)=15sin(30°)/10=0.75. The valid solutions are B≈48.6°, C≈101.4°, c≈19.6 m and B≈131.4°, C≈18.6°, c≈6.4 m, so 2 triangles exist.

Use both correct angle pairs but match b=15 with A=30° when finding c, giving c≈29.4 m and c≈9.6 m instead of the corresponding values.

Keep only principal B≈48.6°, giving C≈101.4° and c≈19.6 m, and report 1 triangle without testing the supplement.

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For the SSA triangle shown, \(A~=~30°\), \(a~=~10~m\), and \(b~=~15~m\). Find every valid triangle, including \(B\), \(C\), \(c\), and the total number of solutions.

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