[skip-to-content]

Exercícios 6.5 Exercícios

1.

Dada a medida do ângulo em graus, determine a medida correspondente em radianos.

  1. \(\displaystyle 45º\)
  2. \(\displaystyle -60º\)
  3. \(\displaystyle 30º\)
  4. \(\displaystyle 345º\)
  5. \(\displaystyle -300º\)
  6. \(\displaystyle 175º\)
2.

Determine o valor exato de cada expressão.

  1. \(\displaystyle \sin\dfrac{2\pi}{3}\)
  2. \(\displaystyle \cos\dfrac{2\pi}{3}\)
  3. \(\displaystyle \tan\dfrac{2\pi}{3}\)
  4. \(\displaystyle \sin\dfrac{7\pi}{6}\)
  5. \(\displaystyle \cos\dfrac{7\pi}{6}\)
  6. \(\displaystyle \sec\dfrac{7\pi}{6}\)
  7. \(\displaystyle \mbox{cossec}\dfrac{7\pi}{6}\)
  8. \(\displaystyle \mbox{cotg}\dfrac{7\pi}{6}\)
  9. \(\displaystyle \tan \dfrac{11\pi}{6}\)
  10. \(\displaystyle \sec \dfrac{11\pi}{6}\)
  11. \(\displaystyle \mbox{cossec}\dfrac{11\pi}{6}\)
  12. \(\displaystyle \mbox{cotg}\dfrac{11\pi}{6}\)
3.

Se \(\cos\theta=-\dfrac{4}{5}\) e \(\pi\leq \theta\leq \dfrac{3\pi}{2}\text{,}\) determine:

  1. \(\displaystyle \sin\theta\)
  2. \(\displaystyle \sec\theta\)
  3. \(\displaystyle \mbox{tg}\hspace{0.05cm}\theta\)
  4. \(\displaystyle \mbox{cossec}\hspace{0.05cm}\theta\)
  5. \(\displaystyle \mbox{cotg}\hspace{0.05cm}\theta\)
4.

Considerando que as funções trigonométricas são periódicas - isto é, \(f(t)=f(t+2\pi k)\text{,}\) em que \(k\) é um inteiro - determine o valor de:

  1. \(\displaystyle \sin\dfrac{43\pi}{4}\)
  2. \(\displaystyle \cos\dfrac{31\pi}{6}\)
  3. \(\displaystyle \mbox{tg}\hspace{0.05cm}\left(\dfrac{-22\pi}{3}\right)\)
  4. \(\displaystyle \mbox{cotg}\hspace{0.05cm}\left(\dfrac{-31\pi}{4}\right)\)
  5. \(\displaystyle \sec\dfrac{71\pi}{6}\)
  6. \(\displaystyle \mbox{cotg}\hspace{0.05cm}\left(\dfrac{-91\pi}{3}\right)\)
5.

Considerando \(\dfrac{5\pi}{6}+\dfrac{\pi}{4}=\dfrac{13\pi}{12}\text{,}\) determine:

  1. \(\displaystyle \sin\dfrac{13\pi}{12}\)
  2. \(\displaystyle \cos\dfrac{13\pi}{12}\)
  3. \(\displaystyle \mbox{tg}\hspace{0.05cm}\dfrac{13\pi}{12}\)
6.

Determine o valor exato de cada expressão, se estiver definido.

  1. \(\displaystyle \arcsin\dfrac{1}{2}\)
  2. \(\displaystyle \arccos\left(-\dfrac{\sqrt{3}}{2}\right)\)
  3. \(\displaystyle \arctan(-1)\)
  4. \(\displaystyle \arcsin\left(-\dfrac{\sqrt{3}}{2}\right)\)
  5. \(\displaystyle \arccos\left(-\dfrac{\sqrt{2}}{2}\right)\)
  6. \(\displaystyle \arctan(-\sqrt{3})\)
  7. \(\displaystyle \arcsin\left(-\dfrac{1}{2}\right)\)
  8. \(\displaystyle \arccos\dfrac{1}{2}\)
  9. \(\displaystyle \arctan\left(\dfrac{\sqrt{3}}{3}\right)\)
  10. \(\displaystyle \arcsin(-1)\)
  11. \(\displaystyle \arccos 1\)
  12. \(\displaystyle \arctan 0\)