Question

How do you write an expression that represents all quadrantal angles?

Answer 1

The expression that represents all quadrantal angkes is
`\phi = n(\pi)/(2)` where
`n` is an integer.

Explanation:

Recall that the quadrantal angles are those coterminal with
`0`,
`(\pi)/(2)`,
`\pi` and
`(3\pi)/(2)`. Now, notice that all of them can be written as integer multiples of
`(\pi)/(2)`:

• 
  `0 = 0\cdot (\pi)/(2)`

• 
  `(\pi)/(2) = 1\cdot (\pi)/(2)`

• 
  `\pi = 2\cdot (\pi)/(2)`

• 
  `(3\pi)/(2) = 3\cdot (\pi)/(2)`

Then, if we add to each one of the listed above angles an integer multiple of
`2\pi` we get

• 
  `0\cdot (\pi)/(2)+2n\pi = (0\pi +4n\pi)/(2) = 4n(\pi)/(2)`

• 
  `1\cdot (\pi)/(2) + 2n\pi = (\pi +4n\pi)/(2) = (4n+1)(\pi)/(2)`

• 
  `2\cdot (\pi)/(2)+ 2n\pi = (2\pi +4n\pi)/(2) = (4n+2)(\pi)/(2)`

• 
  `3\cdot (\pi)/(2)+ 2n\pi = (3\pi +4n\pi)/(2) = (4n+3)(\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(\pi)/(2)` where
`n` is an integer.

Answer 2

These angles coincide with either positive or negative x and y axis. Angles 0 , 90 , 180 , 270 ... So an integer\[\Theta + n \frac{ \pi }{ 2 } , n \in I\] multiplied with pi/2 is the expression.