6 Связь тригонометрических функций одного аргумента

Связь тригонометрических функций одного аргумента

Для решения задач из задания 6 Вам может быть полезно следующее видео:

\(\Longrightarrow\) Как, зная одну тригонометрическую функцию,
найти остальные тригонометрические функции

\(\blacktriangleright\) Примеры. По заданному значению функции найдите значения остальных тригонометрических функций:

1. \(\sin{\alpha}=-\dfrac{20}{29},\,\, \alpha\in \left(\pi;\dfrac{3\pi}{2}\right)\).

   Решение:

   Найдём значение \(\cos{\alpha}\).

   Воспользуемся основным тригонометрическим тождеством: \(\sin^2{\alpha}+\cos^2{\alpha}=1\).

   \[\cos^2{\alpha}=1-\sin^2{\alpha}=1-\left(-\dfrac{20}{29}\right)^2=1-\dfrac{400}{841}=\dfrac{441}{841},\]
   \[\cos{\alpha}=\pm\dfrac{21}{29}.\]

   Так как \(\alpha \in\left(\pi;\dfrac{3\pi}{2}\right)\), то \(\cos{\alpha}<0. \text{ Выбираем } \cos{\alpha}=-\dfrac{21}{29}\).

   Найдём значение \(\tan{\alpha}\).

   Мы знаем значения \(\sin{\alpha}\) и \(\cos{\alpha}\), тогда:

   \[\tan{\alpha}=\dfrac{\sin{\alpha}}{\cos{\alpha}}=\dfrac{-\frac{20}{29}}{-\frac{21}{29}}=\dfrac{20}{21}.\]

   Найдём значение \(\cot{\alpha}\).

   \[\cot{\alpha}=\dfrac{\cos{\alpha}}{\sin{\alpha}}=\dfrac{-\frac{21}{29}}{-\frac{20}{29}}=\dfrac{21}{20}.\]

2. \(\tan{\alpha}=-\dfrac{35}{12},\,\, \alpha\in \left(\dfrac{\pi}{2}; \pi\right)\).

   Решение:

   Найдём \(\cot{\alpha}\).

   \[\tan{\alpha}\cdot \cot{\alpha}=1,\]
   \[\cot{\alpha}=\dfrac{1}{\tan{\alpha}}=\dfrac{1}{-\frac{35}{12}}=-\dfrac{12}{35}.\]

   Найдём \(\cos{\alpha}\).

   \[\tan^2{\alpha}+1=\dfrac{1}{\cos^2{\alpha}},\]
   \[\left(-\dfrac{35}{12}\right)^2+1=\dfrac{1}{\cos^2{\alpha}},\]
   \[\dfrac{1}{\cos^2{\alpha}}=\dfrac{1369}{144},\]
   \[\cos^2{\alpha}=\dfrac{144}{1369},\]
   \[\cos{\alpha}=\pm\dfrac{12}{37}.\]

   Так как \(\alpha \in\left(\dfrac{\pi}{2}; \pi\right)\), то \(\cos{\alpha}<0.\) Выбираем \(\cos{\alpha}=-\dfrac{12}{37}.\)
   Найдём \(\sin{\alpha}\).

   \[\sin^2{\alpha}+\cos^2{\alpha}=1,\]
   \[\sin^2{\alpha}=1-\left(-\dfrac{12}{37}\right)^2=1-\dfrac{144}{1369}=\dfrac{1225}{1369},\]
   \[\sin{\alpha}=\pm \dfrac{35}{37}.\]

   Так как \(\alpha \in\left(\dfrac{\pi}{2}; \pi\right)\), то \(\sin{\alpha}>0.\) Выбираем \(\sin{\alpha}=\dfrac{35}{37}.\)

Задание 6

По заданному значению функции найдите значения остальных тригонометрических функций:

6.1 а) \(\sin t= \dfrac{4}{5}\), \(\dfrac{\pi}{2}<t<\pi\);

б) \(\sin t= \dfrac{5}{13}\), \(0<t<\dfrac{\pi}{2}\);

в) \(\sin t= -0,6\), \(-\dfrac{\pi}{2}<t<0\);

г) \(\sin t= -0,28\), \(\pi<t<\dfrac{3\pi}{2}\).

6.2 а) \(\cos t= 0,8\), \(0<t<\dfrac{\pi}{2}\);

б) \(\cos t= -\dfrac{5}{13}\), \(\dfrac{\pi}{2}<t<\pi\);

в) \(\cos t= 0,6\), \(\dfrac{3\pi}{2}<t<2\pi\);

г) \(\cos t= -\dfrac{24}{25}\), \(\pi<t<\dfrac{3\pi}{2}\).

6.3 а) \(\tan t= \dfrac{3}{4}\), \(0<t<\dfrac{\pi}{2}\);

б) \(\tan t= 2,4\), \(\pi<t<\dfrac{3\pi}{2}\);

в) \(\tan t= -\dfrac{3}{4}\), \(\dfrac{\pi}{2}<t<\pi\);

г) \(\tan t= -\dfrac{5}{12}\), \(\dfrac{3\pi}{2}<t<2\pi\).

6.4 а) \(\cot t= \dfrac{12}{5}\), \(3\pi<t<\dfrac{7\pi}{2}\);

б) \(\cot t= \dfrac{7}{24}\), \(2\pi<t<\dfrac{5\pi}{2}\);

в) \(\cot t= -\dfrac{5}{12}\), \(\dfrac{7\pi}{2}<t<4\pi\);

г) \(\cot t= -\dfrac{8}{15}\), \(\dfrac{5\pi}{2}<t<3\pi\).

Связь тригонометрических функций одного аргумента

Задание 6.1

1. а) — \(\cos t = -\dfrac{3}{5}\); \(\tan t = -\dfrac{4}{3}\); \(\cot t = -\dfrac{3}{4}\).

2. б) — \(\cos t = \dfrac{12}{13}\); \(\tan t = \dfrac{5}{12}\); \(\cot t = \dfrac{12}{5}\).

3. в) — \(\cos t = 0,8\); \(\tan t = -\dfrac{3}{4}\); \(\cot t = -\dfrac{4}{3}\).

4. г) — \(\cos t = -0,96\); \(\tan t = \dfrac{7}{24}\); \(\cot t = \dfrac{24}{7}\).

Задание 5.2

1. а) — \(\sin t = 0,6\); \(\tan t = \dfrac{3}{4}\); \(\cot t = \dfrac{4}{3}\).

2. б) — \(\sin t = \dfrac{12}{13}\); \(\tan t = -\dfrac{12}{5}\); \(\cot t = -\dfrac{5}{12}\).

3. в) — \(\sin t =-0,8\); \(\tan t = -\dfrac{4}{3}\); \(\cot t = -\dfrac{3}{4}\).

4. г) — \(\sin t = -\dfrac{7}{25}\); \(\tan t = \dfrac{7}{24}\); \(\cot t = \dfrac{24}{7}\).

Задание 6.3

1. а) — \(\sin t = \dfrac{3}{5}\); \(\cos t = \dfrac{4}{5}\); \(\cot t = \dfrac{4}{3}\).

2. б) — \(\sin t = -\dfrac{12}{13}\); \(\cos t = -\dfrac{5}{13}\); \(\cot t = \dfrac{5}{12}\).

3. в) — \(\sin t =\dfrac{3}{5}\); \(\cos t = -\dfrac{4}{5}\); \(\cot t = -\dfrac{4}{3}\).

4. г) — \(\sin t = -\dfrac{5}{13}\); \(\cos t = \dfrac{12}{13}\); \(\cot t = -\dfrac{12}{5}\).

Задание 6.4

1. а) — \(\sin t = -\dfrac{5}{13}\); \(\cos t = -\dfrac{12}{13}\); \(\tan t = \dfrac{5}{12}\).

2. б) — \(\sin t = \dfrac{24}{25}\); \(\cos t = \dfrac{7}{25}\); \(\tan t = \dfrac{24}{7}\).

3. в) — \(\sin t =-\dfrac{12}{13}\); \(\cos t = \dfrac{5}{13}\); \(\tan t = -\dfrac{12}{5}\).

4. г) — \(\sin t = \dfrac{15}{17}\); \(\cos t = -\dfrac{8}{17}\); \(\tan t = -\dfrac{15}{8}\).