Question

If i = StartRoot negative 1 EndRoot, what is the value of i 3?

Answer 1

Answer:

i³ = - i

Step-by-step explanation:

Explanation:-

Step(i):-

Given     $i = \sqrt{-1}$

we know that  cos π = -1

                        sin π = 0

now given data

we can write  $i = \sqrt{cos\pi + i sin\pi }$

                      $i =( {cos\pi + i sin\pi })^{\frac{1}{2} }$

step(ii):-

In complex numbers nth roots of a complex number

Z = r(cosθ+i sinθ)

$Z^{\frac{n}{2} } = r^{\frac{n}{2} } (cos\pi +isin\pi )^{\frac{n}{2} }$

$Z^{\frac{n}{2} } = r^{\frac{n}{2} } (cosn\pi/2 }+isinn\pi/2 )$

Step(iii):-

   now       $i =( {cos\frac{\pi }{2} + i sin\frac{\pi }{2} }) }$

                $i^{3} =( {cos\frac{\pi }{2} + i sin\frac{\pi }{2} })^{3} }$

                $i^{3} =( {cos\frac{3\pi }{2} + i sin\frac{3\pi }{2} })}$

By using trigonometric  

              $cos\frac{3\pi }{2} = cos (\pi +\frac{\pi }{2} ) = -cos\frac{\pi }{2} =0$

              $sin\frac{3\pi }{2} = sin (\pi +\frac{\pi }{2} ) = -sin\frac{\pi }{2} = -1$

             $i^{3} =( 0+ i (-1) ) = -i$

Final answer:-

i³ = - i