Answer:
The expression that represents all quadrantal angkes is \(\phi = n\frac{\pi}{2}\) where \(n\) is an integer.
Step-by-step explanation:
Recall that the quadrantal angles are those coterminal with \(0\), \(\frac{\pi}{2}\), \(\pi\) and \(\frac{3\pi}{2}\). Now, notice that all of them can be written as integer multiples of \(\frac{\pi}{2}\):
- \(0 = 0\cdot \frac{\pi}{2}\)
- \(\frac{\pi}{2} = 1\cdot \frac{\pi}{2}\)
- \(\pi = 2\cdot \frac{\pi}{2}\)
- \(\frac{3\pi}{2} = 3\cdot \frac{\pi}{2}\)
Then, if we add to each one of the listed above angles an integer multiple of \(2\pi\) we get
- \(0\cdot \frac{\pi}{2}+2n\pi = \frac{0\pi +4n\pi}{2} = 4n\frac{\pi}{2}\)
- \(1\cdot \frac{\pi}{2} + 2n\pi = \frac{\pi +4n\pi}{2} = (4n+1)\frac{\pi}{2}\)
- \(2\cdot \frac{\pi}{2}+ 2n\pi = \frac{2\pi +4n\pi}{2} = (4n+2)\frac{\pi}{2}\)
- \(3\cdot \frac{\pi}{2}+ 2n\pi = \frac{3\pi +4n\pi}{2} = (4n+3)\frac{\pi}{2}\)
But the numbers of the form \(4n\), \(4n+1\), \(4n+2\) and \(4n+3\) where \(n\) is an integer, encompass all integers. Thus, all quadrantal angles can be written as
\(\phi = n\frac{\pi}{2}\) where \(n\) is an integer.
