Реши примеры с тригонометрическими функциями

2. 1) $6 \sin 90^\circ - 3 \cos 180^\circ = 6 \cdot 1 - 3 \cdot (-1) = 6 + 3 = 9$. 2) $2 \cos 0^\circ + \tan 0^\circ = 2 \cdot 1 + 0 = 2$. 3) $\sin^2 50^\circ + \cos^2 50^\circ = 1$ (основное тригонометрическое тождество). 4) $\sin^2 20^\circ + \cos^2 160^\circ = \sin^2 20^\circ + \cos^2 (180^\circ - 20^\circ) = \sin^2 20^\circ + (-\cos 20^\circ)^2 = \sin^2 20^\circ + \cos^2 20^\circ = 1$. 3. 1) Дано $\sin \alpha = \frac{2}{3}$ и $0^\circ < \alpha < 90^\circ$. Тогда $\cos \alpha = \sqrt{1 - \sin^2 \alpha} = \sqrt{1 - (\frac{2}{3})^2} = \sqrt{1 - \frac{4}{9}} = \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3}$. 2) Дано $\cos \alpha = -\frac{1}{5}$. Тогда $\sin \alpha = \pm \sqrt{1 - \cos^2 \alpha} = \pm \sqrt{1 - (-\frac{1}{5})^2} = \pm \sqrt{1 - \frac{1}{25}} = \pm \sqrt{\frac{24}{25}} = \pm \frac{2\sqrt{6}}{5}$. 3) Дано $\sin \alpha = \frac{5}{6}$. Тогда $\cos \alpha = \pm \sqrt{1 - \sin^2 \alpha} = \pm \sqrt{1 - (\frac{5}{6})^2} = \pm \sqrt{1 - \frac{25}{36}} = \pm \sqrt{\frac{11}{36}} = \pm \frac{\sqrt{11}}{6}$. 4. 1) $\cos 102^\circ \cdot \cot 92^\circ < 0$, так как $\cos 102^\circ < 0$ и $\cot 92^\circ < 0$. 2) $\sin 0^\circ \cdot \cos 28^\circ \cdot \tan 82^\circ = 0 \cdot \cos 28^\circ \cdot \tan 82^\circ = 0$. 5. 1) $\cos 120^\circ \cdot \sin 135^\circ \cdot \cot 150^\circ = -\frac{1}{2} \cdot \frac{\sqrt{2}}{2} \cdot (-\sqrt{3}) = \frac{\sqrt{6}}{4}$. 2) $4 \tan^2 120^\circ + 4 \sin^2 120^\circ - 3 \cos 90^\circ \cdot \cot 100^\circ = 4 \cdot (-\sqrt{3})^2 + 4 \cdot (\frac{\sqrt{3}}{2})^2 - 3 \cdot 0 \cdot \cot 100^\circ = 4 \cdot 3 + 4 \cdot \frac{3}{4} - 0 = 12 + 3 = 15$. 6. 1) $\frac{\cos 123^\circ - \tan 141^\circ}{\cos 57^\circ - \tan 39^\circ} = \frac{\cos (90^\circ + 33^\circ) - \tan (90^\circ + 51^\circ)}{\cos 57^\circ - \tan 39^\circ} = \frac{-\sin 33^\circ - (-\cot 51^\circ)}{\cos 57^\circ - \tan 39^\circ} = \frac{-\sin 33^\circ + \cot 51^\circ}{\cos 57^\circ - \tan 39^\circ} = \frac{-\sin 33^\circ + \frac{1}{\tan 51^\circ}}{\cos 57^\circ - \tan 39^\circ} = \frac{-\sin 33^\circ + \frac{1}{\tan (90^\circ - 39^\circ)}}{\cos 57^\circ - \tan 39^\circ} = \frac{-\sin 33^\circ + \frac{1}{\cot 39^\circ}}{\cos 57^\circ - \tan 39^\circ} = \frac{-\sin 33^\circ + \tan 39^\circ}{\cos 57^\circ - \tan 39^\circ} = \frac{-\cos 57^\circ + \tan 39^\circ}{\cos 57^\circ - \tan 39^\circ} = -1$. 2) $\frac{\sin 18^\circ}{\sin 162^\circ} + \frac{\cot 103^\circ}{\cot 77^\circ} = \frac{\sin 18^\circ}{\sin (180^\circ - 18^\circ)} + \frac{\cot (90^\circ + 13^\circ)}{\cot 77^\circ} = \frac{\sin 18^\circ}{\sin 18^\circ} + \frac{-\tan 13^\circ}{\cot 77^\circ} = 1 + \frac{-\tan 13^\circ}{\cot (90^\circ - 13^\circ)} = 1 + \frac{-\tan 13^\circ}{\tan 13^\circ} = 1 - 1 = 0$.