Б 1) tg$\frac{13\pi}{6}$ - sin$\frac{5\pi}{3}$; 2) ctg$\frac{5\pi}{3}$ + cos$\frac{7\pi}{6}$; 3) sin$\frac{20\pi}{3}$ + 2cos$\frac{13\pi}{3}$; 4) tg(-510°) - ctg 1020°; 5) cos$\frac{7\pi}{4}$ + tg$-25\pi$; 6) ctg$\frac{13\pi}{6}$ + sin$\frac{8\pi}{3}$; 7) $\sqrt{(2\sin 45° - 1)^2} + \sqrt{(1 - \cos 45°)^2} - 4\sin 495°$; 8) 2$\cos$(-135°)$\sqrt{2}$ - 2$\cos$ 30°; 9) $\sin$$\alpha + \beta$$\sin$$\alpha – \beta$ при $\alpha$ = 45°, $\beta$ = 15°; 10) tg$2\alpha -\beta$ + $\cos$ $\alpha$ $\operatorname{ctg}$$6\alpha + 6\beta$ при $\alpha$ = 20°, $\beta$ = -5°; 11) $\log$_2 $\sin$$\frac{\pi}{4}$ + $\log$_1 $\cos$$\frac{\pi}{2}$; 12) $\log$_3 $\cos$ 30° - $\log$_9 $\sin$^2 30°.

Question

Б 1) tg$\frac{13\pi}{6}$ - sin$\frac{5\pi}{3}$; 2) ctg$\frac{5\pi}{3}$ + cos$\frac{7\pi}{6}$; 3) sin$\frac{20\pi}{3}$ + 2cos$\frac{13\pi}{3}$; 4) tg(-510°) - ctg 1020°; 5) cos$\frac{7\pi}{4}$ + tg$-25\pi$; 6) ctg$\frac{13\pi}{6}$ + sin$\frac{8\pi}{3}$; 7) $\sqrt{(2\sin 45° - 1)^2} + \sqrt{(1 - \cos 45°)^2} - 4\sin 495°$; 8) 2$\cos$(-135°)$\sqrt{2}$ - 2$\cos$ 30°; 9) $\sin$$\alpha + \beta$$\sin$$\alpha – \beta$ при $\alpha$ = 45°, $\beta$ = 15°; 10) tg$2\alpha -\beta$ + $\cos$ $\alpha$ $\operatorname{ctg}$$6\alpha + 6\beta$ при $\alpha$ = 20°, $\beta$ = -5°; 11) $\log$_2 $\sin$$\frac{\pi}{4}$ + $\log$_1 $\cos$$\frac{\pi}{2}$; 12) $\log$_3 $\cos$ 30° - $\log$_9 $\sin$^2 30°.

$Фото задания: Б 1) tg$\frac{13\pi}{6}$ - sin$\frac{5\pi}{3}$; 2) ctg$\frac{5\pi}{3}$ + cos$\frac{7\pi}{6}$; 3)…$ Фото задания

Accepted Answer

Решение:

1) $\operatorname{tg}\frac{13\pi}{6} - \sin\frac{5\pi}{3} = \operatorname{tg}(\frac{12\pi}{6} + \frac{\pi}{6}) - \sin(\frac{6\pi}{3} - \frac{\pi}{3}) = \operatorname{tg}(\frac{\pi}{6}) - \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{3} - \frac{\sqrt{3}}{2} = \frac{2\sqrt{3} - 3\sqrt{3}}{6} = -\frac{\sqrt{3}}{6}$

2) $\operatorname{ctg}\frac{5\pi}{3} + \cos\frac{7\pi}{6} = \operatorname{ctg}(2\pi - \frac{\pi}{3}) + \cos(\pi + \frac{\pi}{6}) = -\operatorname{ctg}\frac{\pi}{3} - \cos\frac{\pi}{6} = -\frac{\sqrt{3}}{3} - \frac{\sqrt{3}}{2} = \frac{-2\sqrt{3} - 3\sqrt{3}}{6} = -\frac{5\sqrt{3}}{6}$

3) $\sin\frac{20\pi}{3} + 2\cos\frac{13\pi}{3} = \sin(6\pi + \frac{2\pi}{3}) + 2\cos(4\pi + \frac{\pi}{3}) = \sin\frac{2\pi}{3} + 2\cos\frac{\pi}{3} = \frac{\sqrt{3}}{2} + 2 \cdot \frac{1}{2} = \frac{\sqrt{3}}{2} + 1 = \frac{\sqrt{3}+2}{2}$

4) $\operatorname{tg}(-510°) - \operatorname{ctg}(1020°) = \operatorname{tg}(-510°+2\cdot 360°) - \operatorname{ctg}(1020°-2\cdot 360°) = \operatorname{tg}(210°) - \operatorname{ctg}(300°) = \operatorname{tg}(180°+30°) - \operatorname{ctg}(360°-60°) = \operatorname{tg}30° - (-\operatorname{ctg}60°) = \frac{1}{\sqrt{3}} + \frac{1}{\sqrt{3}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$

5) $\cos(-\frac{7\pi}{4}) + \operatorname{tg}(-25\pi) = \cos(\frac{7\pi}{4}) + \operatorname{tg}(-25\pi) = \cos(2\pi - \frac{\pi}{4}) + 0 = \cos\frac{\pi}{4} = \frac{\sqrt{2}}{2}$

6) $\operatorname{ctg}(-\frac{13\pi}{6}) + \sin(\frac{8\pi}{3}) = -\operatorname{ctg}(\frac{13\pi}{6}) + \sin(\frac{8\pi}{3}) = -\operatorname{ctg}(2\pi + \frac{\pi}{6}) + \sin(2\pi + \frac{2\pi}{3}) = -\operatorname{ctg}\frac{\pi}{6} + \sin\frac{2\pi}{3} = -\sqrt{3} + \frac{\sqrt{3}}{2} = -\frac{\sqrt{3}}{2}$

7) $\sqrt{(2\sin 45° - 1)^2} + \sqrt{(1 - \cos 45°)^2} - 4\sin 495°$

$\sqrt{(2\frac{\sqrt{2}}{2} - 1)^2} + \sqrt{(1 - \frac{\sqrt{2}}{2})^2} - 4\sin(495°-360°)$

$\sqrt{(\sqrt{2} - 1)^2} + \sqrt{(\frac{2-\sqrt{2}}{2})^2} - 4\sin(135°)$

$|\sqrt{2}-1| + |\frac{2-\sqrt{2}}{2}| - 4\frac{\sqrt{2}}{2}$

$(\sqrt{2}-1) + \frac{2-\sqrt{2}}{2} - 2\sqrt{2}$

$\sqrt{2}-1 + 1 - \frac{\sqrt{2}}{2} - 2\sqrt{2}$

$-\frac{\sqrt{2}}{2} - \sqrt{2} = -\frac{3\sqrt{2}}{2}$

8) $2\cos(-135°)\sqrt{2} - 2\cos 30° = 2\cos(180°-45°)\sqrt{2} - 2\frac{\sqrt{3}}{2} = -2\cos 45°\sqrt{2} - \sqrt{3} = -2\frac{\sqrt{2}}{2}\sqrt{2} - \sqrt{3} = -\sqrt{2}\sqrt{2} - \sqrt{3} = -2 - \sqrt{3}$

9) $\sin(\alpha + \beta)\sin(\alpha – \beta) = \sin^2 \alpha - \sin^2 \beta$

При $\alpha = 45°, \beta = 15°$: $\sin^2 45° - \sin^2 15° = (\frac{\sqrt{2}}{2})^2 - (\frac{\sqrt{6}-\sqrt{2}}{4})^2 = \frac{2}{4} - \frac{6 - 2\sqrt{12} + 2}{16} = \frac{1}{2} - \frac{8 - 4\sqrt{3}}{16} = \frac{1}{2} - \frac{2-\sqrt{3}}{4} = \frac{2 - (2-\sqrt{3})}{4} = \frac{\sqrt{3}}{4}$

10) $\operatorname{tg}(2\alpha -\beta) + \cos \alpha \operatorname{ctg}(6\alpha + 6\beta)$

При $\alpha = 20°, \beta = -5°$: $\alpha = 20°, \beta = -5°$

$2\alpha - \beta = 2(20°) - (-5°) = 40° + 5° = 45°$

$6\alpha + 6\beta = 6(20°) + 6(-5°) = 120° - 30° = 90°$

$\operatorname{tg}(45°) + \cos(20°) \operatorname{ctg}(90°) = 1 + \cos(20°) \cdot 0 = 1$

11) $\log_2 \sin\(\frac{\pi}{4}$ + $\log$_1 $\cos$$\frac{\pi}{2}$ = $\log$_2 $\frac{\sqrt{2}}{2}$ + $\log$_1 0\)

Логарифм по основанию 1 не определен, так же как и логарифм от 0.

12) $\log_3 \cos 30° - \log_9 \sin^2 30°$

$\log_3 \frac{\sqrt{3}}{2} - \log_{3^2} (\frac{1}{2})^2 = \log_3 \frac{\sqrt{3}}{2} - \frac{1}{2} \log_3 \frac{1}{4}$

$\log_3 \frac{\sqrt{3}}{2} - \frac{1}{2} \log_3 2^{-2} = \log_3 \frac{\sqrt{3}}{2} - \frac{1}{2} (-2) \log_3 2 = \log_3 \frac{\sqrt{3}}{2} + \log_3 2 = \log_3 (\frac{\sqrt{3}}{2} \cdot 2) = \log_3 \sqrt{3} = \log_3 3^{1/2} = \frac{1}{2}$

Ответ: 1) -$\frac{\sqrt{3}}{6}$; 2) -$\frac{5\sqrt{3}}{6}$; 3) $\frac{\sqrt{3}+2}{2}$; 4) $\frac{2\sqrt{3}}{3}$; 5) $\frac{\sqrt{2}}{2}$; 6) -$\frac{\sqrt{3}}{2}$; 7) -$\frac{3\sqrt{2}}{2}$; 8) -2-$\sqrt{3}$; 9) $\frac{\sqrt{3}}{4}$; 10) 1; 11) Не определено; 12) $\frac{1}{2}$.