Trigonometric Identities

  1. Determine the exact value, expressed in terms of radicals if necessary, for each of the following outputs of trigonometric functions.
    \(\displaystyle \cos(15°)\)
    \(\displaystyle \sin(15°)\)
    \(\displaystyle \tan(15°)\)
    \(\displaystyle \cos(7.5°)\)
    \(\displaystyle \sin(7.5°)\)
    \(\displaystyle \tan(7.5°)\)
    \(\displaystyle \cos(22.5°)\)
    \(\displaystyle \cos(11.25°)\)
    \(\displaystyle \cos(5.625°)\)
    \(\displaystyle \cos(2.8125°)\)
  2. Write each of the following expression exclusively in terms of cosine.
    \(\displaystyle \sin(\theta) \)
    \(\displaystyle \tan(\theta) \)
    \(\displaystyle \sec(\theta)-\sin^2(\theta) \)
    \(\displaystyle \sin\Bigl(\frac{1}{2}\theta\Bigr) + \sin(2\theta) \)
  3. Given that \(\cos(36°) = \frac{\sqrt{5}+1}{4},\) determine the exact value, expressed in terms of radicals if necessary, for each of the following outputs of trigonometric functions.
    \(\displaystyle \sin(36°) \)
    \(\displaystyle \cos(72°) \)
    \(\displaystyle \cos(18°) \)
    \(\displaystyle \cos(24°) \)
    \(\displaystyle \cos(9°) \)
    \(\displaystyle \cos(69°) \)