Question

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

Answer 1

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.

Answer 2

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.