5 Вычислите: a) \(cos 315^\circ + sin 210^\circ + tg 420^\circ\) б) \(sin \frac{13\pi}{6} - cos \frac{11\pi}{6} + ctg \frac{11\pi}{4}\)

Краткое пояснение:

Решение:

• а) \(cos 315^\circ + sin 210^\circ + tg 420^\circ\)
  • \(cos 315^\circ = cos (360^\circ - 45^\circ) = cos 45^\circ = \frac{\sqrt{2}}{2}\)
  • \(sin 210^\circ = sin (180^\circ + 30^\circ) = -sin 30^\circ = -\frac{1}{2}\)
  • \(tg 420^\circ = tg (360^\circ + 60^\circ) = tg 60^\circ = \sqrt{3}\)

  Следовательно, \(cos 315^\circ + sin 210^\circ + tg 420^\circ = \frac{\sqrt{2}}{2} - \frac{1}{2} + \sqrt{3}\)

• б) \(sin \frac{13\pi}{6} - cos \frac{11\pi}{6} + ctg \frac{11\pi}{4}\)
  • \(sin \frac{13\pi}{6} = sin (2\pi + \frac{\pi}{6}) = sin \frac{\pi}{6} = \frac{1}{2}\)
  • \(cos \frac{11\pi}{6} = cos (2\pi - \frac{\pi}{6}) = cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\)
  • \(ctg \frac{11\pi}{4} = ctg (2\pi + \frac{3\pi}{4}) = ctg \frac{3\pi}{4} = -1\)

  Следовательно, \(sin \frac{13\pi}{6} - cos \frac{11\pi}{6} + ctg \frac{11\pi}{4} = \frac{1}{2} - \frac{\sqrt{3}}{2} - 1 = -\frac{1}{2} - \frac{\sqrt{3}}{2}\)

Ответ: а) \(\frac{\sqrt{2}}{2} - \frac{1}{2} + \sqrt{3}\); б) \(-\frac{1}{2} - \frac{\sqrt{3}}{2}\)