What is the polar form of #(16,-4#?

Answer 1

#( 4sqrt(17) , -0,245 )#

The quickest method is to use the formula:

#(x , y ) = ( sqrt(x^2+y^2), arctan(y/x))#

Below is an alternative method.

#x=rcostheta#
#y=rsintheta#
#:.#
#16=rcostheta# [ 1 ]
#-4=rsintheta# [ 2 ]

Squaring both equations:

#256=r^2cos^2(theta)#
#16=r^2sin^2(theta)#

Adding both equations:

#272=r^2cos^2(theta)+r^2sin^2(theta)#
#272=r^2(cos^2(theta)+sin^2(theta))#
#272=r^2(1)=>r=+-sqrt(272)=+-4sqrt(17)#

We only need the positive root, negative radii can exist, and a point can be represented by many different polar coordinates, unlike Cartesian coordinates which are unique.

#theta=arctan(y/x)=arctan(-4/16)=arctan(-1/4)=-0.24498#

Polar coordinate:

#( 4sqrt(17) , -0,245 )#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To find the polar form of the complex number (16 - 4i), where (i) is the imaginary unit, follow these steps:

  1. Convert the rectangular form to polar form using the formulas: [ r = \sqrt{x^2 + y^2} ] [ \theta = \arctan\left(\frac{y}{x}\right) ]

  2. Substitute the given values (x = 16) and (y = -4) into the formulas.

  3. Calculate (r) and (\theta).

  4. Write the polar form as (r(\cos(\theta) + i \sin(\theta))).

Let's calculate:

[ r = \sqrt{16^2 + (-4)^2} = \sqrt{256 + 16} = \sqrt{272} = 4\sqrt{17} ]

[ \theta = \arctan\left(\frac{-4}{16}\right) = \arctan\left(-\frac{1}{4}\right) ]

Using a calculator, we find ( \theta \approx -0.24498 ) radians.

So, the polar form of (16 - 4i) is ( 4\sqrt{17} \left( \cos(-0.24498) + i \sin(-0.24498) \right) ).

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7