“Unit Circle” Trigonometry

Exercises

  1. What are the exact values, possibly expressed in terms of radicals (i.e. not just decimal approximations), of each of the following outputs of a trigonometric function? Note that some arguments are expressed in degree measure (°) while others are expressed in radian measure.
    \(\displaystyle \cos\bigl(30°\bigr)\)
    \(\displaystyle \cos\bigl(45°\bigr)\)
    \(\displaystyle \cos\bigl(60°\bigr)\)
    \(\displaystyle \cos\bigl(90°\bigr)\)
    \(\displaystyle \cos\bigl(180°\bigr)\)
    \(\displaystyle \sin\bigl(180°\bigr)\)
    \(\displaystyle \sin\bigl(30°\bigr)\)
    \(\displaystyle \sin\bigl(45°\bigr)\)
    \(\displaystyle \tan\bigl(45°\bigr)\)
    \(\displaystyle \tan\bigl(30°\bigr)\)
    \(\displaystyle \tan\bigl(60°\bigr)\)
    \(\displaystyle \cos\bigl(300°\bigr)\)
    \(\displaystyle \sin\bigl(-30°\bigr)\)
    \(\displaystyle \cos\bigl(225°\bigr)\)
    \(\displaystyle \sin\bigl(-150°\bigr)\)
    \(\displaystyle \tan\bigl(315°\bigr)\)
    \(\displaystyle \cos\bigl(135°\bigr)\)
    \(\displaystyle \cos\bigl(-135°\bigr)\)
    \(\displaystyle \tan\bigl(-135°\bigr)\)
    \(\displaystyle \sec\bigl(30°\bigr)\)
    \(\displaystyle \csc\bigl(45°\bigr)\)
    \(\displaystyle \cot\bigl(60°\bigr)\)
    \(\displaystyle \sec\bigl(300°\bigr)\)
    \(\displaystyle \sec\bigl(-135°\bigr)\)
    \(\displaystyle \csc\bigl(240°\bigr)\)
    \(\displaystyle \cot\bigl(-45°\bigr)\)
    \(\displaystyle \cos\biggl(12960°\)
    \(\displaystyle \sin\biggl(\frac{3\pi}{2}\biggr)\)
    \(\displaystyle \cos\biggl(\frac{\pi}{6}\biggr)\)
    \(\displaystyle \sin\biggl(\frac{5\pi}{6}\biggr)\)
    \(\displaystyle \cos\biggl(\frac{7\pi}{3}\biggr)\)
    \(\displaystyle \sin\biggl(\frac{-3\pi}{4}\biggr)\)
    \(\displaystyle \cos\biggl(\frac{-2\pi}{3}\biggr)\)
    \(\displaystyle \tan\biggl(\frac{\pi}{3}\biggr)\)
    \(\displaystyle \cot\biggl(\frac{\pi}{6}\biggr)\)
    \(\displaystyle \cos\biggl(\frac{-2\pi}{3}\biggr)\)
    \(\displaystyle \sin\biggl(\frac{170\pi}{3}\biggr)\)
  2. What’s the smallest positive value \(\theta\) such that \(\cos\bigl(\theta\bigr) = \frac{1}{2}?\)
  3. What’s the smallest positive value \(\theta\) such that \(\cos\bigl(\theta\bigr) = -\frac{1}{2}?\)
  4. What’s the smallest positive value \(\theta\) such that \(\tan\bigl(\theta\bigr) = \sqrt{3}?\)
  5. Suppose that \(\sin\bigl(\theta\bigr) = \frac{3}{4}\) for some value of \(\theta.\) What are all the possible positive values of \(\cos\bigl(\theta\bigr)?\) What are all the possible positive values of \(\tan\bigl(\theta\bigr)?\)
  6. Suppose that \(\sec\bigl(\theta\bigr) = 4\) for some value of \(\theta.\) and that \(\tan\bigl(\theta\bigr) \lt 0.\) What must the value of \(\sin\bigl(\theta\bigr)\) be?
  7. Suppose that \(\tan\bigl(\theta\bigr) = 5\) for some value of \(\theta.\) What are all the possible positive values of \(\sin\bigl(\theta\bigr)?\) What are all the possible positive values of \(\cos\bigl(\theta\bigr)?\)
  8. Given that \(\cos\bigl(\theta\bigr)= \frac{5}{7}\) what value(s) might \(\sin\bigl(\theta\bigr)\) have? What value(s) might \(\tan\bigl(\theta\bigr)\) have?
  9. Given that \(\sin\bigl(\theta\bigr) = \frac{1}{3}\) and that the terminal point of \(\theta\) is in quadrant II (where \(x\) is negative and \(y\) is positive) what must the value of \(\cos\bigl(\theta\bigr)\) be? What must the value of \(\csc\bigl(\theta\bigr)\) be? What must the value of \(\tan\bigl(\theta\bigr)\) be?
  10. Does there exist a value of \(\theta\) such that \(\cos\bigl(\theta\bigr) = 2?\) If so, approximately what is it? If not, why not?
  11. 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?
  12. Sketch a picture of the unit circle, and on the picture shade in a sector of the circle with an internal angle measuring 3 radians. Don’t just convert 3 radians to degrees; remember what radian measure means.

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.
  3. How do you determine the exact value of \(\cos\bigl(\frac{\pi}{5}\bigr)\) and \(\sin\bigl(\frac{\pi}{5}\bigr)?\)