How do you write the complex number in trigonometric form $- 7 i$ $-7i$ ?

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Answer 1 · 2018-01-06T10:22:39

The answer is $= 7 (cos (- \frac{π}{2}) + i sin (- \frac{π}{2})) = 7 e^{- \frac{1}{2} i π}$ $=7(cos(-pi/2)+isin(-pi/2))=7e^(-1/2ipi)$

Any complex number $z = a + i b$ $z=a+ib$ can be represented as

$z = r (cos θ + i sin θ)$ $z=r(costheta+isintheta)$

Where,

$r = | | z | | = \sqrt{a^{2} + b^{2}}$ $r=||z||=sqrt(a^2+b^2)$

$cos θ = \frac{a}{| | z | |}$ $costheta=a/(||z||)$

and

$sin θ = \frac{b}{| | z | |}$ $sintheta=b/||z||$

Here, we have

$z = 0 - 7 i$ $z=0-7i$

$| | z | | = \sqrt{{(0)}^{2} + {(- 7)}^{2}} = 7$ $||z||=sqrt((0)^2+(-7)^2)=7$

$z = 7 ((\frac{0}{7}) + (- \frac{7}{7}) i)$ $z=7((0/7)+(-7/7)i)$

$cos θ = 0$ $costheta=0$ and $sin θ = - 1$ $sin theta=-1$

Therefore,

$θ = - \frac{π}{2}$ $theta =-pi/2$ , $[mod 2pi]$

So,

$z=7(cos(-pi/2)+isin(-pi/2))$

How do you write the complex number in trigonometric form -7i−7i-7i?