Question

Solve the triangle.

A = 52°, b = 10, c = 7


A.) a ≈ 12, C ≈ 47.9, B ≈ 80.1

B.) a ≈ 7.9, C ≈ 43.9, B ≈ 84.1

C.) No triangles possible

D.) a ≈ 12, C ≈ 43.9, B ≈ 84.1

Answer 1

a ≈ 7.9, B ≈ 83.9°, C ≈ 44.1° — best approximated by Choice B).

Explanation:

You want the solution to the triangle with A = 52°, b = 10, c = 7.

Law of cosines

When given two sides and the angle between, the law of cosines can help you find the length of the third side.

a² = b² +c² -2bc·cos(A)

a² = 10² +7² -2·10·7·cos(52°) = 149 -140·cos(52°) ≈ 62.8074

a ≈ √62.8074 ≈ 7.9 . . . . . . identifies choice B as the best choice

Law of sines

Now that we have an angle and its opposite side, we can find the other angles using the Law of Sines:

sin(B) = b·sin(A)/a = 10·sin(52°)/7.92511 ≈

B ≈ 83.89° ≈ 83.9°

C = 180° -52° -83.9° = 44.1°

The solution is a ≈ 7.9, B ≈ 83.9°, C ≈ 44.1°.

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Additional comment

The discrepancy between these answers and the ones offered with the problem points to inappropriate rounding (or some other error) during computation of the answer key.

A triangle is always possible when an angle between two sides is given.