Question

Express the complex number in trigonometric form. -5i

Answer 1

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°)

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))

Answer 2

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

Explanation:

Look at the picture.

The trigonometric form of a complex number:

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

where:

`|z|=√(a^2+b^2)\\\\\cos\alpha=(a)/(|z|)\\\\\sin\alpha=(b)/(|z|)`

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

Substitute:

`|z|=√(0^2+(-5)^2)=√(0+25)=√(25)=5\\\\\cos\theta=(0)/(5)=0\\\\\sin\theta=(-5)/(5)=-1`

Therefore

`\theta=(3\pi)/(2)`

Finally:

`-5i=5\left(\cos(3\pi)/(2)+i\sin(3\pi)/(2)\right)`