§ 6. Синус и косинус. Тангенс и котангенс
Вычислите \(\sin t\), \(\cos t\) и \(\text{tg } t\), если:
6.1.
а) \(t = 0\)
\[\sin 0 = 0\] \[\cos 0 = 1\] \[\text{tg } 0 = \frac{\sin 0}{\cos 0} = \frac{0}{1} = 0\]
б) \(t = \frac{\pi}{2}\)
\[\sin \frac{\pi}{2} = 1\] \[\cos \frac{\pi}{2} = 0\] \[\text{tg } \frac{\pi}{2} = \frac{\sin \frac{\pi}{2}}{\cos \frac{\pi}{2}} = \frac{1}{0}\] Тангенс не определён, так как деление на ноль невозможно.
в) \(t = \frac{3\pi}{2}\)
\[\sin \frac{3\pi}{2} = -1\] \[\cos \frac{3\pi}{2} = 0\] \[\text{tg } \frac{3\pi}{2} = \frac{\sin \frac{3\pi}{2}}{\cos \frac{3\pi}{2}} = \frac{-1}{0}\] Тангенс не определён.
г) \(t = \pi\)
\[\sin \pi = 0\] \[\cos \pi = -1\] \[\text{tg } \pi = \frac{\sin \pi}{\cos \pi} = \frac{0}{-1} = 0\]
6.2.
а) \(t = -2\pi\)
Используем свойство периодичности: \(\sin(t + 2\pi k) = \sin t\), \(\cos(t + 2\pi k) = \cos t\). \[\sin (-2\pi) = \sin (0 - 2\pi) = \sin 0 = 0\] \[\cos (-2\pi) = \cos (0 - 2\pi) = \cos 0 = 1\] \[\text{tg } (-2\pi) = \frac{\sin (-2\pi)}{\cos (-2\pi)} = \frac{0}{1} = 0\]
б) \(t = -\frac{\pi}{2}\)
Используем свойства нечётности синуса и чётности косинуса: \(\sin(-t) = -\sin t\), \(\cos(-t) = \cos t\). \[\sin \left(-\frac{\pi}{2}\right) = -\sin \frac{\pi}{2} = -1\] \[\cos \left(-\frac{\pi}{2}\right) = \cos \frac{\pi}{2} = 0\] \[\text{tg } \left(-\frac{\pi}{2}\right) = \frac{\sin \left(-\frac{\pi}{2}\right)}{\cos \left(-\frac{\pi}{2}\right)} = \frac{-1}{0}\] Тангенс не определён.
в) \(t = -\frac{3\pi}{2}\)
\[\sin \left(-\frac{3\pi}{2}\right) = -\sin \frac{3\pi}{2} = -(-1) = 1\] \[\cos \left(-\frac{3\pi}{2}\right) = \cos \frac{3\pi}{2} = 0\] \[\text{tg } \left(-\frac{3\pi}{2}\right) = \frac{\sin \left(-\frac{3\pi}{2}\right)}{\cos \left(-\frac{3\pi}{2}\right)} = \frac{1}{0}\] Тангенс не определён.
г) \(t = -\pi\)
\[\sin (-\pi) = -\sin \pi = -0 = 0\] \[\cos (-\pi) = \cos \pi = -1\] \[\text{tg } (-\pi) = \frac{\sin (-\pi)}{\cos (-\pi)} = \frac{0}{-1} = 0\]
6.3.
а) \(t = \frac{5\pi}{6}\)
Угол \(\frac{5\pi}{6}\) находится во II четверти. \[\sin \frac{5\pi}{6} = \sin \left(\pi - \frac{\pi}{6}\right) = \sin \frac{\pi}{6} = \frac{1}{2}\] \[\cos \frac{5\pi}{6} = \cos \left(\pi - \frac{\pi}{6}\right) = -\cos \frac{\pi}{6} = -\frac{\sqrt{3}}{2}\] \[\text{tg } \frac{5\pi}{6} = \frac{\sin \frac{5\pi}{6}}{\cos \frac{5\pi}{6}} = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}\]
б) \(t = \frac{5\pi}{4}\)
Угол \(\frac{5\pi}{4}\) находится в III четверти. \[\sin \frac{5\pi}{4} = \sin \left(\pi + \frac{\pi}{4}\right) = -\sin \frac{\pi}{4} = -\frac{\sqrt{2}}{2}\] \[\cos \frac{5\pi}{4} = \cos \left(\pi + \frac{\pi}{4}\right) = -\cos \frac{\pi}{4} = -\frac{\sqrt{2}}{2}\] \[\text{tg } \frac{5\pi}{4} = \frac{\sin \frac{5\pi}{4}}{\cos \frac{5\pi}{4}} = \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = 1\]
в) \(t = \frac{7\pi}{6}\)
Угол \(\frac{7\pi}{6}\) находится в III четверти. \[\sin \frac{7\pi}{6} = \sin \left(\pi + \frac{\pi}{6}\right) = -\sin \frac{\pi}{6} = -\frac{1}{2}\] \[\cos \frac{7\pi}{6} = \cos \left(\pi + \frac{\pi}{6}\right) = -\cos \frac{\pi}{6} = -\frac{\sqrt{3}}{2}\] \[\text{tg } \frac{7\pi}{6} = \frac{\sin \frac{7\pi}{6}}{\cos \frac{7\pi}{6}} = \frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\]
г) \(t = \frac{7\pi}{4}\)
Угол \(\frac{7\pi}{4}\) находится в IV четверти. \[\sin \frac{7\pi}{4} = \sin \left(2\pi - \frac{\pi}{4}\right) = -\sin \frac{\pi}{4} = -\frac{\sqrt{2}}{2}\] \[\cos \frac{7\pi}{4} = \cos \left(2\pi - \frac{\pi}{4}\right) = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}\] \[\text{tg } \frac{7\pi}{4} = \frac{\sin \frac{7\pi}{4}}{\cos \frac{7\pi}{4}} = \frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = -1\]
6.4.
а) \(t = -\frac{7\pi}{4}\)
\[\sin \left(-\frac{7\pi}{4}\right) = -\sin \frac{7\pi}{4} = - \left(-\frac{\sqrt{2}}{2}\right) = \frac{\sqrt{2}}{2}\] \[\cos \left(-\frac{7\pi}{4}\right) = \cos \frac{7\pi}{4} = \frac{\sqrt{2}}{2}\] \[\text{tg } \left(-\frac{7\pi}{4}\right) = \frac{\sin \left(-\frac{7\pi}{4}\right)}{\cos \left(-\frac{7\pi}{4}\right)} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1\]
б) \(t = -\frac{4\pi}{3}\)
\[\sin \left(-\frac{4\pi}{3}\right) = -\sin \frac{4\pi}{3} = -\sin \left(\pi + \frac{\pi}{3}\right) = - \left(-\sin \frac{\pi}{3}\right) = \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}\] \[\cos \left(-\frac{4\pi}{3}\right) = \cos \frac{4\pi}{3} = \cos \left(\pi + \frac{\pi}{3}\right) = -\cos \frac{\pi}{3} = -\frac{1}{2}\] \[\text{tg } \left(-\frac{4\pi}{3}\right) = \frac{\sin \left(-\frac{4\pi}{3}\right)}{\cos \left(-\frac{4\pi}{3}\right)} = \frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = -\sqrt{3}\]
в) \(t = -\frac{5\pi}{6}\)
\[\sin \left(-\frac{5\pi}{6}\right) = -\sin \frac{5\pi}{6} = -\sin \left(\pi - \frac{\pi}{6}\right) = -\sin \frac{\pi}{6} = -\frac{1}{2}\] \[\cos \left(-\frac{5\pi}{6}\right) = \cos \frac{5\pi}{6} = \cos \left(\pi - \frac{\pi}{6}\right) = -\cos \frac{\pi}{6} = -\frac{\sqrt{3}}{2}\] \[\text{tg } \left(-\frac{5\pi}{6}\right) = \frac{\sin \left(-\frac{5\pi}{6}\right)}{\cos \left(-\frac{5\pi}{6}\right)} = \frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\]
г) \(t = -\frac{5\pi}{3}\)
\[\sin \left(-\frac{5\pi}{3}\right) = -\sin \frac{5\pi}{3} = -\sin \left(2\pi - \frac{\pi}{3}\right) = - \left(-\sin \frac{\pi}{3}\right) = \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}\] \[\cos \left(-\frac{5\pi}{3}\right) = \cos \frac{5\pi}{3} = \cos \left(2\pi - \frac{\pi}{3}\right) = \cos \frac{\pi}{3} = \frac{1}{2}\] \[\text{tg } \left(-\frac{5\pi}{3}\right) = \frac{\sin \left(-\frac{5\pi}{3}\right)}{\cos \left(-\frac{5\pi}{3}\right)} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}\]
6.5.
а) \(t = \frac{13\pi}{6}\)
\[\sin \frac{13\pi}{6} = \sin \left(2\pi + \frac{\pi}{6}\right) = \sin \frac{\pi}{6} = \frac{1}{2}\] \[\cos \frac{13\pi}{6} = \cos \left(2\pi + \frac{\pi}{6}\right) = \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\] \[\text{tg } \frac{13\pi}{6} = \frac{\sin \frac{13\pi}{6}}{\cos \frac{13\pi}{6}} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\]
б) \(t = -\frac{8\pi}{3}\)
\[\sin \left(-\frac{8\pi}{3}\right) = -\sin \frac{8\pi}{3} = -\sin \left(2\pi + \frac{2\pi}{3}\right) = -\sin \frac{2\pi}{3} = -\sin \left(\pi - \frac{\pi}{3}\right) = -\sin \frac{\pi}{3} = -\frac{\sqrt{3}}{2}\] \[\cos \left(-\frac{8\pi}{3}\right) = \cos \frac{8\pi}{3} = \cos \left(2\pi + \frac{2\pi}{3}\right) = \cos \frac{2\pi}{3} = \cos \left(\pi - \frac{\pi}{3}\right) = -\cos \frac{\pi}{3} = -\frac{1}{2}\] \[\text{tg } \left(-\frac{8\pi}{3}\right) = \frac{\sin \left(-\frac{8\pi}{3}\right)}{\cos \left(-\frac{8\pi}{3}\right)} = \frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = \sqrt{3}\]
в) \(t = \frac{23\pi}{6}\)
\[\sin \frac{23\pi}{6} = \sin \left(4\pi - \frac{\pi}{6}\right) = -\sin \frac{\pi}{6} = -\frac{1}{2}\] \[\cos \frac{