Question

Express the complex number in trigonometric form. -5i

Answer 1

Answer:

$\large\boxed{-5i=5\left(\cos\dfrac{3\pi}{2}+i\sin\dfrac{3\pi}{2}\right)}$

Step-by-step explanation:

Look at the picture.

The trigonometric form of a complex number:

$z=|z|(\cos\alpha+i\sin\alpha)$

where:

$|z|=\sqrt{a^2+b^2}\\\\\cos\alpha=\dfrac{a}{|z|}\\\\\sin\alpha=\dfrac{b}{|z|}$

We have the complex number z = - 5i → z = 0 + (-5)i → a = 0, b = -5.

Substitute:

$|z|=\sqrt{0^2+(-5)^2}=\sqrt{0+25}=\sqrt{25}=5\\\\\cos\theta=\dfrac{0}{5}=0\\\\\sin\theta=\dfrac{-5}{5}=-1$

Therefore

$\theta=\dfrac{3\pi}{2}$

Finally:

$-5i=5\left(\cos\dfrac{3\pi}{2}+i\sin\dfrac{3\pi}{2}\right)$

Answer 2

Answer:

Choices:

A) 5(cos 270° + i sin 270°)

B) 5(cos 180° + i sin 180°)

C) 5(cos 90° + i sin 90°)

D) 5(cos 0° + i sin 0°)

Step-by-step explanation:

-5i can be written as 0 + (-5)i

It is in the form a+bi where a = 0 and b =-5

So the point (a,b) is (0,-5)

The distance from the origin to this point is 5 units, therefore r = 5. This is the magnitude.  

The angle is 270 degrees as shown in the attached image. You start on the positive x axis and rotate until you reach the point (0,-5)

This is why the answer is choice A) 5(cos(270) + i*sin(270))