Problem

For the triangle shown, \(K~=~20\), \(a~=~8\), and \(b~=~10\). Find the maximum possible area, solve the area formula for every included angle \(C\), and state the number of solutions.

Source triangle labeled with area 20, adjacent sides 8 and 10, unknown included angle C, and perpendicular height h.

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What this problem is really about

First compare the target area with the maximum possible area for the two fixed sides, which occurs when the included angle has sine one. If the target is attainable, isolate the sine of the angle and test both the inverse-sine value and its supplement. Count distinct angles strictly between zero and one hundred eighty degrees.

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Find C=30° and 150° but count them as one solution because their sines are equal.

Use only principal C=arcsin(1/2)=30° and report 1 solution without testing 150°.

The maximum area is ab/2=40. Since sin(C)=2K/(ab)=1/2, C=30° or 150°, so 2 included angles are valid.

Omit the factor 2: sin(C)=K/(ab)=1/4, so C≈14.5° or 165.5° and report 2 solutions.

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For the triangle shown, \(K~=~20\), \(a~=~8\), and \(b~=~10\). Find the maximum possible area, solve the area formula for every included angle \(C\), and state the number of solutions.

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