“Unit Circle” Trigonometry

  1. What are the exact values, possibly expressed in terms of radicals, of each of the following outputs of a trigonometric function?
    \(\displaystyle \cos\biggl(\frac{0\pi}{12}\biggr)\)
    \(\displaystyle \sin\biggl(\frac{3\pi}{2}\biggr)\)
    \(\displaystyle \cos\biggl(\frac{\pi}{6}\biggr)\)
    \(\displaystyle \sin\biggl(\frac{5\pi}{6}\biggr)\)
    \(\displaystyle \cos\biggl(\frac{7\pi}{3}\biggr)\)
    \(\displaystyle \sin\biggl(\frac{-3\pi}{4}\biggr)\)
    \(\displaystyle \cos\biggl(\frac{-2\pi}{3}\biggr)\)
    \(\displaystyle \sin\biggl(\frac{170\pi}{3}\biggr)\)
  2. Sketch a picture of the unit circle, and on the picture draw a dot  •  at the terminal point on the unit circle corresponding to an arclength of \(t=3.\) What are the \(x\)- and \(y\)-coordinates of this point? Round each coordinate to three decimal places.
  3. What are the exact values, possibly expressed in terms of radicals, of each of the following outputs of a trigonometric function?
    \(\displaystyle \tan\biggl(\frac{0\pi}{12}\biggr)\)
    \(\displaystyle \csc\biggl(\frac{3\pi}{2}\biggr)\)
    \(\displaystyle \tan\biggl(\frac{\pi}{6}\biggr)\)
    \(\displaystyle \sec\biggl(\frac{5\pi}{6}\biggr)\)
    \(\displaystyle \sec\biggl(\frac{7\pi}{3}\biggr)\)
    \(\displaystyle \cot\biggl(\frac{-3\pi}{4}\biggr)\)
    \(\displaystyle \csc\biggl(\frac{-2\pi}{3}\biggr)\)
    \(\displaystyle \tan\biggl(\frac{170\pi}{3}\biggr)\)
  4. Given that \(\sin(t)= \frac{1}{3}\) and that \(t\) is in quadrant II (where \(x\) is negative and \(y\) is positive) what must the value of \(\cos(t)\) be? What must the value of \(\csc(t)\) be? What must the value of \(\tan(t)\) be?
  5. Given that \(\cos(t)= \frac{5}{7}\) what value(s) might \(\sin(t)\) have? What value(s) might \(\tan(t)\) have?
  6. The domains of sine and cosine each consist of all real numbers. I.e. \(\sin(t)\) and \(\cos(t)\) can be evaluated at any input value \(t.\) This is not true for tangent, secant, cosecant, and cosecant, however. Because each of them is defined as a ratio (quotient) we must be careful to avoid division by zero. For each of these four functions, determine the values of \(t\) that must be excluded from their domains.