Figure 3.6.2 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

Regarding these as linear equations in the unknowns cosA, cosB, and cosC, use Cramer’s rule to derive the law of cosines by solving for

Thus

a² = b² + c² - 2bc cosA.

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

c cos B + bcos C = a c cos A + a cos C = b a cos B + bcos A = c.