485. Найти значение выражения: 1) sin 73° cos 17° + cos 73° sin 17°; 2) sin 73° cos 13° − cos 73° sin 13°; 3) sin(5π/12) cos(π/12) + sin(π/12) cos(5π/12); 4) sin(7π/12) cos(π/12) − sin(π/12) cos(7π/12).

Question

Вопрос:

Accepted Answer

Используем формулы синуса суммы и разности: \(\sin A\cos B+\cos A\sin B=\sin(A+B)\), \(\sin A\cos B-\cos A\sin B=\sin(A-B)\).

\(\sin73^\circ\cos17^\circ+\cos73^\circ\sin17^\circ=\sin(73^\circ+17^\circ)=\sin90^\circ=1\).
\(\sin73^\circ\cos13^\circ-\cos73^\circ\sin13^\circ=\sin(73^\circ-13^\circ)=\sin60^\circ=\frac{\sqrt3}{2}\).
\(\sin\frac{5\pi}{12}\cos\frac{\pi}{12}+\sin\frac{\pi}{12}\cos\frac{5\pi}{12}=\sin\left(\frac{5\pi}{12}+\frac{\pi}{12}\right)=\sin\frac{\pi}{2}=1\).
\(\sin\frac{7\pi}{12}\cos\frac{\pi}{12}-\sin\frac{\pi}{12}\cos\frac{7\pi}{12}=\sin\left(\frac{7\pi}{12}-\frac{\pi}{12}\right)=\sin\frac{\pi}{2}=1\).

Ответ: 1) 1; 2) \(\frac{\sqrt3}{2}\); 3) 1; 4) 1.