| Задача 1. Вычислите \({\text{tg}}\frac{\pi }{4} + \sqrt 3 {\text{ctg}}\frac{\pi }{3}\)
|
| Задача 2. Вычислите \(\sqrt 3 {\text{tg}}\frac{\pi }{3} — \frac{{\sqrt 3 }}{2}{\text{ctg}}\frac{\pi }{6}\)
|
| Задача 3. Вычислите \({\text{tg}}\frac{{3\pi }}{4} — \sqrt {27} {\text{ctg}}\frac{{2\pi }}{3}\)
|
| Задача 4. Вычислите \(\frac{1}{{\sqrt 3 }}\,\,{\text{ctg}}\frac{{7\pi }}{6} — \sqrt {12} \,{\text{tg}}\frac{{4\pi }}{3}\)
|
| Задача 5. Вычислите \(\frac{1}{5}{\text{tg}}\frac{{7\pi }}{4} + \frac{{\sqrt {27} }}{5}{\text{ctg}}\frac{{11\pi }}{6}\)
|
| Задача 6. Вычислите \( — \sqrt 3 {\text{tg}}\left( { — \frac{{2\pi }}{3}} \right) + {\text{ctg}}\left( { — \frac{{5\pi }}{4}} \right)\)
|
| Задача 7. Вычислите \(\frac{2}{{\sqrt 3 }}{\text{ctg}}\left( { — \frac{{7\pi }}{6}} \right) — \frac{{\sqrt 3 }}{2}{\text{tg}}\left( { — \frac{{5\pi }}{3}} \right)\)
|
| Задача 8. Вычислите \(\sqrt 3 {\text{tg}}\frac{{8\pi }}{3} — {\text{ctg}}\frac{{17\pi }}{4}\)
|
| Задача 9. Вычислите \(2\left( {{\text{tg}}\left( { — \frac{{15\pi }}{4}} \right) + {\text{ctg}}\left( { — \frac{{17\pi }}{4}} \right)} \right)\)
|
| Задача 10. Вычислите \(\frac{{\sqrt {12} }}{5}\left( {{\text{ctg}}\frac{{19\pi }}{3} — {\text{tg}}\frac{{23\pi }}{6}} \right)\)
|
| Задача 11. Вычислите \(\frac{1}{{\sqrt 3 }}{\text{tg}}\frac{{29\pi }}{3} — \frac{{\sqrt 3 }}{2}{\text{ctg}}\frac{{34\pi }}{3}\)
|
| Задача 12. Вычислите \(\frac{{\sqrt {12} }}{5}\left( {{\text{tg}}\frac{{59\pi }}{6} — {\text{ctg}}\frac{{40\pi }}{3}} \right)\)
|
| Задача 13. Вычислите \(\sqrt {108} \left( {{\text{tg}}{{390}^ \circ } — {\text{2ctg}}{{480}^ \circ }} \right)\)
|
| Задача 14. Вычислите \({\text{tg}}{45^ \circ } \cdot {\text{ctg}}{135^ \circ }\)
|
| Задача 15. Вычислите \(\left( {{\text{tg}}{{720}^ \circ } + {\text{ctg}}{{405}^ \circ }} \right) \cdot \sin {330^ \circ }\)
|
| Задача 16. Вычислите \(\left( {{\text{tg}}{{585}^ \circ } + 3{\text{ctg}}{{135}^ \circ }} \right) \cdot \cos \left( { — {{240}^ \circ }} \right)\)
|
| Задача 17. Вычислите \({\text{tg}}\left( { — {{80}^ \circ }} \right) + {\text{tg}}\left( { — {{70}^ \circ }} \right) + {\text{tg}}\left( { — {{60}^ \circ }} \right) + \ldots + {\text{tg}}\,{\text{6}}{{\text{0}}^ \circ } + {\text{tg}}\,7{{\text{0}}^ \circ } + {\text{tg}}\,8{{\text{0}}^ \circ }\)
|
| Задача 18. Вычислите \({\text{ctg}}\,{10^ \circ }\,\,{\text{ctg}}\,{20^ \circ }\,\,{\text{ctg}}\,{30^ \circ } \cdot \ldots \cdot {\text{ctg}}\,{160^ \circ }\,\,{\text{ctg}}\,{170^ \circ }\)
|
Задача 19. Расположите в порядке возрастания числа: \({\text{tg}}\frac{\pi }{5};\,\,\,\,{\text{tg}}\frac{{2\pi }}{5};\,\,\,{\text{tg}}\frac{{4\pi }}{5}\)
| 1) \({\text{tg}}\frac{\pi }{5};\,\,\,\,{\text{tg}}\frac{{2\pi }}{5};\,\,\,{\text{tg}}\frac{{4\pi }}{5}\) |
2) \({\text{tg}}\frac{\pi }{5};\,\,\,\,{\text{tg}}\frac{{4\pi }}{5};\,\,\,{\text{tg}}\frac{{2\pi }}{5}\) |
| 3) \({\text{tg}}\frac{{2\pi }}{5};\,\,\,\,{\text{tg}}\frac{\pi }{5};\,\,\,{\text{tg}}\frac{{4\pi }}{5}\) |
4) \({\text{tg}}\frac{{2\pi }}{5};\,\,\,\,{\text{tg}}\frac{{4\pi }}{5};\,\,\,{\text{tg}}\frac{\pi }{5}\) |
| 5) \({\text{tg}}\frac{{4\pi }}{5};\,\,\,\,{\text{tg}}\frac{\pi }{5};\,\,\,{\text{tg}}\frac{{2\pi }}{5}\) |
6) \({\text{tg}}\frac{{4\pi }}{5};\,\,\,\,{\text{tg}}\frac{{2\pi }}{5};\,\,\,{\text{tg}}\frac{\pi }{5}\) |
|
Задача 20. Расположите в порядке возрастания числа: \({\text{tg}}\left( { — \frac{\pi }{7}} \right);\,\,\,\,{\text{tg}}\frac{{5\pi }}{7};\,\,\,{\text{tg}}\frac{{8\pi }}{7}\)
| 1) \({\text{tg}}\left( { — \frac{\pi }{7}} \right);\,\,\,\,{\text{tg}}\frac{{5\pi }}{7};\,\,\,{\text{tg}}\frac{{8\pi }}{7}\) |
2) \({\text{tg}}\left( { — \frac{\pi }{7}} \right);\,\,\,\,{\text{tg}}\frac{{8\pi }}{7};\,\,\,{\text{tg}}\frac{{5\pi }}{7}\) |
| 3) \({\text{tg}}\frac{{5\pi }}{7};\,\,\,\,{\text{tg}}\left( { — \frac{\pi }{7}} \right);\,\,\,{\text{tg}}\frac{{8\pi }}{7}\) |
4) \({\text{tg}}\frac{{5\pi }}{7};\,\,\,\,{\text{tg}}\frac{{8\pi }}{7};\,\,\,\,{\text{tg}}\left( { — \frac{\pi }{7}} \right)\) |
| 5) \({\text{tg}}\frac{{8\pi }}{7};\,\,\,\,{\text{tg}}\left( { — \frac{\pi }{7}} \right);\,\,\,{\text{tg}}\frac{{5\pi }}{7}\) |
6) \({\text{tg}}\frac{{8\pi }}{7};\,\,\,\,{\text{tg}}\frac{{5\pi }}{7};\,\,\,\,{\text{tg}}\left( { — \frac{\pi }{7}} \right)\) |
|
Задача 21. Расположите в порядке возрастания числа: \({\text{ctg}}\frac{{2\pi }}{5};\,\,\,\,{\text{ctg}}\frac{{3\pi }}{5};\,\,\,{\text{ctg}}\frac{{6\pi }}{5}\)
| 1) \({\text{ctg}}\frac{{2\pi }}{5};\,\,\,\,{\text{ctg}}\frac{{3\pi }}{5};\,\,\,{\text{ctg}}\frac{{6\pi }}{5}\) |
2) \({\text{ctg}}\frac{{2\pi }}{5};\,\,\,\,{\text{ctg}}\frac{{6\pi }}{5};\,\,\,{\text{ctg}}\frac{{3\pi }}{5}\) |
| 3) \({\text{ctg}}\frac{{3\pi }}{5};\,\,\,\,{\text{ctg}}\frac{{2\pi }}{5};\,\,\,{\text{ctg}}\frac{{6\pi }}{5}\) |
4) \({\text{ctg}}\frac{{3\pi }}{5};\,\,\,\,{\text{ctg}}\frac{{6\pi }}{5};\,\,\,{\text{ctg}}\frac{{2\pi }}{5}\) |
| 5) \({\text{ctg}}\frac{{6\pi }}{5};\,\,\,\,{\text{ctg}}\frac{{3\pi }}{5};\,\,\,{\text{ctg}}\frac{{2\pi }}{5}\) |
6) \({\text{ctg}}\frac{{6\pi }}{5};\,\,\,\,{\text{ctg}}\frac{{2\pi }}{5};\,\,\,{\text{ctg}}\frac{{3\pi }}{5}\) |
|
Задача 22. Расположите в порядке возрастания: \({\text{ctg}}\left( { — \frac{{2\pi }}{9}} \right);\,\,\,\,{\text{ctg}}\frac{{5\pi }}{9};\,\,\,{\text{ctg}}\frac{{8\pi }}{9}\)
| 1) \({\text{ctg}}\left( { — \frac{{2\pi }}{9}} \right);\,\,\,\,{\text{ctg}}\frac{{5\pi }}{9};\,\,\,{\text{ctg}}\frac{{8\pi }}{9}\) |
2) \({\text{ctg}}\left( { — \frac{{2\pi }}{9}} \right);\,\,\,\,{\text{ctg}}\frac{{8\pi }}{9};\,\,\,{\text{ctg}}\frac{{5\pi }}{9}\) |
| 3) \(\,{\text{ctg}}\frac{{5\pi }}{9};\,\,\,{\text{ctg}}\frac{{8\pi }}{9}{\text{;}}\,\,\,\,{\text{ctg}}\left( { — \frac{{2\pi }}{9}} \right)\) |
4) \(\,{\text{ctg}}\frac{{5\pi }}{9};\,\,\,{\text{ctg}}\left( { — \frac{{2\pi }}{9}} \right);\,\,\,\,{\text{ctg}}\frac{{8\pi }}{9}\) |
| 5) \(\,{\text{ctg}}\frac{{8\pi }}{9};\,\,\,{\text{ctg}}\frac{{5\pi }}{9}{\text{;}}\,\,\,\,{\text{ctg}}\left( { — \frac{{2\pi }}{9}} \right)\) |
6) \(\,{\text{ctg}}\frac{{8\pi }}{9};\,\,\,{\text{ctg}}\left( { — \frac{{2\pi }}{9}} \right);\,\,\,\,{\text{ctg}}\frac{{5\pi }}{9}\) |
|
Задача 23. Расположите в порядке убывания числа: \({\text{tg}}\frac{{3\pi }}{7};\,\,\,\,{\text{tg}}\frac{{8\pi }}{7};\,\,\,{\text{ctg}}\frac{{4\pi }}{5}\)
| 1) \({\text{tg}}\frac{{3\pi }}{7};\,\,\,\,\,{\text{ctg}}\frac{{4\pi }}{5};\,\,\,\,{\text{tg}}\frac{{8\pi }}{7}\) |
2) \({\text{tg}}\frac{{3\pi }}{7};\,\,\,\,{\text{tg}}\frac{{8\pi }}{7};\,\,\,{\text{ctg}}\frac{{4\pi }}{5}\) |
| 3) \({\text{tg}}\frac{{8\pi }}{7};\,\,\,\,\,{\text{ctg}}\frac{{4\pi }}{5};\,\,\,\,{\text{tg}}\frac{{3\pi }}{7}\) |
4) \({\text{tg}}\frac{{8\pi }}{7};\,\,\,\,\,{\text{tg}}\frac{{3\pi }}{7};\,\,\,\,{\text{ctg}}\frac{{4\pi }}{5}\) |
| 5) \({\text{ctg}}\frac{{4\pi }}{5};\,\,\,\,{\text{tg}}\frac{{8\pi }}{7};\,\,\,\,\,\,{\text{tg}}\frac{{3\pi }}{7}\) |
6) \({\text{ctg}}\frac{{4\pi }}{5};\,\,\,\,{\text{tg}}\frac{{3\pi }}{7};\,\,\,\,\,\,{\text{tg}}\frac{{8\pi }}{7}\) |
|
Задача 24. Расположите в порядке убывания числа: \(\operatorname{c} {\text{tg}}\frac{\pi }{9};\,\,\,\,{\text{tg}}\frac{{5\pi }}{9};\,\,\,\,\sin \frac{\pi }{9}\)
| 1) \(\operatorname{c} {\text{tg}}\frac{\pi }{9};\,\,\,\,{\text{tg}}\frac{{5\pi }}{9};\,\,\,\,\sin \frac{\pi }{9}\) |
2) \(\operatorname{c} {\text{tg}}\frac{\pi }{9};\,\,\,\,\sin \frac{\pi }{9};\,\,\,\,{\text{tg}}\frac{{5\pi }}{9}\) |
| 3) \({\text{tg}}\frac{{5\pi }}{9};\,\,\,\,{\text{ctg}}\frac{\pi }{9};\,\,\,\,\sin \frac{\pi }{9}\) |
4) \({\text{tg}}\frac{{5\pi }}{9};\,\,\,\,\sin \frac{\pi }{9};\,\,\,\,{\text{ctg}}\frac{\pi }{9}\) |
| 5) \(\sin \frac{\pi }{9};\,\,\,\,{\text{ctg}}\frac{\pi }{9};\,\,\,\,{\text{tg}}\frac{{5\pi }}{9}\) |
6) \(\sin \frac{\pi }{9};\,\,\,\,{\text{tg}}\frac{{5\pi }}{9};\,\,\,\,{\text{ctg}}\frac{\pi }{9}\) |
|
| Задача 25. Найдите значение выражения \(\sqrt 3 \left( {{\text{tg}}\,x + {\text{ctg}}\,2x} \right)\), если \(x = \frac{\pi }{3}\)
|
| Задача 26. Найдите значение выражения \({\text{3t}}{{\text{g}}^2}2x — {\text{ct}}{{\text{g}}^2}2x\), если \(x = \frac{\pi }{{12}}\)
|
| Задача 27. Найдите значение выражения \(\frac{{{\text{tg}}\,x + {\text{ctg}}\,x}}{{{{\left( {\sin x + \cos x} \right)}^2}}}\), если \(x = \frac{{5\pi }}{4}\)
|
| Задача 28. Найдите значение выражения \(\frac{{\sin 3x + \sqrt 3 \,{\text{ctg}}\,3x}}{{\sqrt 3 \,{\text{ctg}}\,6x — {\text{t}}{{\text{g}}^2}6x}}\), если \(x = \frac{\pi }{{18}}\)
|