“Unit Circle” Trigonometry

Exercises

  1. What’s the smallest positive value \(t\) such that \(\cos(t) = \frac{1}{2}?\)
  2. What’s the smallest positive value \(t\) such that \(\cos(t) = -\frac{1}{2}?\)
  3. What’s the smallest positive value \(t\) such that \(\tan(t) = \sqrt{3}?\)
  4. Suppose that \(\sin(t) = \frac{3}{4}\) for some value of \(t.\) What are all the possible positive values of \(\cos(t)?\) What are all the possible positive values of \(\tan(t)?\)
  5. Suppose that \(\sec(t) = 4\) for some value of \(t\) and that \(\tan(t) \lt 0.\) What must the value of \(\sin(t)\) be?
  6. Suppose that \(\tan(t) = 5\) for some value of \(t.\) What are all the possible positive values of \(\sin(t)?\) What are all the possible positive values of \(\cos(t)?\)
  7. Does there exist a value of \(t\) such that \(\cos(t) = 2?\) If so, approximately what is it? If not, why not?
  8. Name any point \((x,y)\) where the graphs of sine and cosine intersect. What is the significance of this point in the context of the unit circle?
  9. What are the exact values, possibly expressed in terms of radicals, of each of the following outputs of a trigonometric function?
    \(\displaystyle \cos\biggl(\frac{0\pi}{12}\biggr)\)
    \(\displaystyle \sin\biggl(\frac{3\pi}{2}\biggr)\)
    \(\displaystyle \cos\biggl(\frac{\pi}{6}\biggr)\)
    \(\displaystyle \sin\biggl(\frac{5\pi}{6}\biggr)\)
    \(\displaystyle \cos\biggl(\frac{7\pi}{3}\biggr)\)
    \(\displaystyle \sin\biggl(\frac{-3\pi}{4}\biggr)\)
    \(\displaystyle \cos\biggl(\frac{-2\pi}{3}\biggr)\)
    \(\displaystyle \sin\biggl(\frac{170\pi}{3}\biggr)\)
  10. 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.
  11. What are the exact values, possibly expressed in terms of radicals, of each of the following outputs of a trigonometric function?
    \(\displaystyle \tan\biggl(\frac{0\pi}{12}\biggr)\)
    \(\displaystyle \csc\biggl(\frac{3\pi}{2}\biggr)\)
    \(\displaystyle \tan\biggl(\frac{\pi}{6}\biggr)\)
    \(\displaystyle \sec\biggl(\frac{5\pi}{6}\biggr)\)
    \(\displaystyle \sec\biggl(\frac{7\pi}{3}\biggr)\)
    \(\displaystyle \cot\biggl(\frac{-3\pi}{4}\biggr)\)
    \(\displaystyle \csc\biggl(\frac{-2\pi}{3}\biggr)\)
    \(\displaystyle \tan\biggl(\frac{170\pi}{3}\biggr)\)
  12. 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?
  13. Given that \(\cos(t)= \frac{5}{7}\) what value(s) might \(\sin(t)\) have? What value(s) might \(\tan(t)\) have?
  14. 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.

Problems & Challenges

  1. Define the chord function \(\operatorname{crd}\) to take as input an angle \(\theta\) and return as \(\operatorname{crd}(\theta)\) the length of the line segment (the chord) between the points \((1,0)\) and \(\bigl(\cos(\theta), \sin(\theta)\bigr)\) on the unit circle. The chord function is useful sometimes, but it isn’t particularly novel; it can be written as a simple transformation of the sine function. Figure out how to express a formula for \(\operatorname{crd}(\theta)\) as a composite of sine with linear functions.
  2. a unit circle featuring colored line segments that represent the lengths of many different trigonometric functions Typically only the sine and cosines functions are taught to be lengths in the context of the “unit circle,” while other trig functions are defined algebraically in terms of sine and cosine. However, many these functions can also be thought of as lengths on the “unit circle.” The diagram here shows some of these functions, as well as a few antiquated functions like versine and exsecant, as lengths. Recalling some basic geometric knowledge, confirm that the definitions of tangent and secant and cotangent and cosecant in terms of sine and cosine agree with the lengths of the line segments labelled with those trig functions in this diagram.