Figure shows an acute triangle with angles A, B, and C and opposite sides a, b, and c. By...

Figure shows an acute triangle with angles A, B, and C and opposite sides a, b, and c. By dropping a perpendicular from each vertex to the opposite side, derive the equations

c cos B + b cos C = a

c cos A + b cos C = b

c cos B + b cos A = c

Regarding these as linear equations in the unknowns cos A, cos B, and cos C, use Cramer’s rule to derive the law of cosines by solving for

Thus

a2 = b2 + c2 − 2bc cos A.

Note that the case A = π/2 (90°) reduces to the Pythagorean theorem.