1. Вычислите: а) 3 cos 60° − 2 sin 30° + 6 ctg 60° − 2 ctg 30°; б) sin(−30°) + cos(−60°) − 2 tg(−30°) ctg(−60°); в) 5 sin(−45°) + 5 cos(−45°) − √3 tg(−30°) + sin(−30°).

Question

Вопрос:

Accepted Answer

а) Подставим значения тригонометрических функций:

\(3\cdot\frac12-2\cdot\frac12+6\cdot\frac{1}{\sqrt3}-2\sqrt3=\frac32-1+2\sqrt3-2\sqrt3=\frac12.\)

б) \(\sin(-30°)=-\frac12\), \(\cos(-60°)=\frac12\), \( g(-30°)=-\frac{1}{\sqrt3}\), \(\ctg(-60°)=-\frac{1}{\sqrt3}\).

\(-\frac12+\frac12-2\cdot\left(-\frac{1}{\sqrt3}\right)\cdot\left(-\frac{1}{\sqrt3}\right)=-\frac23.\)

в) \(\sin(-45°)=-\frac{\sqrt2}{2}\), \(\cos(-45°)=\frac{\sqrt2}{2}\), \( g(-30°)=-\frac{1}{\sqrt3}\), \(\sin(-30°)=-\frac12\).

\(5\cdot\left(-\frac{\sqrt2}{2}\right)+5\cdot\frac{\sqrt2}{2}-\sqrt3\cdot\left(-\frac{1}{\sqrt3}\right)-\frac12=\frac12.\)

Ответ: а) \(\frac12\); б) \(-\frac23\); в) \(\frac12\).