Exercise 4.6.4 solution

We want to find the principle angle whose cosine is the same as the cosine of negative \(-\dfrac{\pi}{3}\).

Principle angles for cosine lie between \(0\) and \(\pi\) inclusive. The angle \(-\dfrac{\pi}{3}\) lies in quadrant IV, so it has a positive cosine. The principle angle having the same cosine as \(-\dfrac{\pi}{3}\) lies in quadrant I.

  1. Sketch the \(x\) and \(y\) axes and draw a unit circle.
  2. Draw the terminal side of the angle \(-\dfrac{\pi}{3}\).
  3. Draw a vertical line through the point where the terminal side of of the angle \(-\dfrac{\pi}{3}\) intersects the unit circle.
  4. That vertical line will intersect the circle at a point \(P\) in the first quadrant.
  5. Draw a line through \(P\) and the origin.
  6. That line is the terminal side of the angle we are looking for.
  7. It is the angle \(\dfrac{\pi}{3}\).

The principle angle whose cosine is the same as the cosine of \(-\dfrac{\pi}{3}\) is \(\dfrac{\pi}{3}\).