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