9. ∠A = 23°10′, ∠C = 41°15′, c = 10. Найти cos A, cos C, ∠B, cos B, a, b.

Сумма углов треугольника равна \(180^\circ\): \(\angle B=180^\circ-23^\circ10'-41^\circ15'=115^\circ35'\).

Косинусы углов: \(\cos A=\cos(23^\circ10')\approx0.9192\), \(\cos C=\cos(41^\circ15')\approx0.7522\), \(\cos B=\cos(115^\circ35')\approx-0.4325\).

По теореме синусов: \(a=c\sin A/\sin C=10\sin(23^\circ10')/\sin(41^\circ15')\approx5.95\); \(b=c\sin B/\sin C=10\sin(115^\circ35')/\sin(41^\circ15')\approx13.65\).

Ответ: cos A ≈ 0.9192; cos C ≈ 0.7522; ∠B = 115°35′; cos B ≈ −0.4325; a ≈ 5.95; b ≈ 13.65.