Convert the complex number z = 12 ( cos 90 ° + i ・ sin 90 ° ) z=12\left(\cos90\degree+i・\sin90\degree\right) z = 12 ( cos 90° + i ・ sin 90° ) from polar to rectangular form.

this problem, we are asked to take this complex number in polar form and convert it to rectangular form. Now recall there is a straightforward way to do this. What you first wanna do is take whatever this r value is out in front and distribute it into the parenthesis. So let's go ahead and do this step. If we do this, we're going to have 12 times the cosine of 90 degrees. That's going to be plus I times 12 times the sine of 90 degrees. Now from here, all you need to do is figure out what 12 cos 90 is and what 12 sine 90 is and that will allow you to solve the problem. There are a bunch of different ways you could do this. You could use a calculator or a unit circle, but another way that you could do this is realize that 90 degrees is actually one of the angles that we're very familiar with. If you were to look at a unit circle, this would tell you that we're right here on the y axis, meaning that the cosine value there is going to be 0, and the sine is going to be 1. Now again, you could actually verify this with a calculator if you wanted to be sure, but if you do this correctly you should get 12 times 0 plus I times 12 times 1. Now, 12 times 0 is just going to be 0, and we're going to have this plus 12 times 1 times I, which is just 12 I, and then this whole thing will simplify to just 12 I. So this right here is our complex number in rectangular form, and that is how you can solve this problem.

11. Graphing Complex Numbers

Polar Form of Complex Numbers

Trigonometry

11. Graphing Complex Numbers

Polar Form of Complex Numbers

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Multiple Choice

Convert the complex number $z=12\left(\cos90\degree+i・\sin90\degree\right)$ from polar to rectangular form.

$z=12i$

$z=12$

$z=1$

$z=i$

Verified step by step guidance

Understand that the complex number is given in polar form as z = r(cos(θ) + i sin(θ)), where r is the magnitude and θ is the angle.

Identify the values from the given polar form: r = 12 and θ = 90°.

Recall that cos(90°) = 0 and sin(90°) = 1. Substitute these values into the polar form expression.

Calculate the rectangular form using the formula: z = r(cos(θ) + i sin(θ)) = 12(0 + i * 1).

Simplify the expression to find the rectangular form: z = 12i.

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Struggling with Trigonometry?

Convert the complex number z=12(cos90°+i・sin90°)z=12\left(\cos90\degree+i・\sin90\degree\right)z=12(cos90°+i・sin90°) from polar to rectangular form.

Watch next

Polar Form of Complex Numbers practice set

Convert the complex number $z=12\left(\cos90\degree+i・\sin90\degree\right)$ from polar to rectangular form.