What is the polar form of #(42,77)#?

Question

Charles Anctil · Accepted Answer

#sqrt(7693)cis(1.071)#

Quick way of doing this: Use the Pol button on ur calculator and enter the coordinates.

If #z# is the complex number, Finding modulus: #|z|=sqrt(42^2+77^2)=sqrt(7693)#

Finding argument: Plot the point on an Argand diagram. This is important to ensure that you write the principal argument. We can see that the complex number is in the first quadrant, so no adjustments need to be made, but be wary when the point is in the 3rd/4th quadrants.

Arg#(z)=tan^-1(77/42)=1.071# radians or #61°23'#

Placing this in polar form, #z=|z|cisarg(z)=sqrt(7693)cis1.071#

Samantha Adkison · Answer

undefined The polar form of the point (42, 77) in Cartesian coordinates is $ (r, \theta) $, where $ r $ is the distance from the origin to the point and $ \theta $ is the angle measured counterclockwise from the positive x-axis to the line segment connecting the origin to the point.

To find $ r $, use the distance formula: 
$$ r = \sqrt{x^2 + y^2} $$

To find $ \theta $, use the arctangent function:
$$ \theta = \arctan\left(\frac{y}{x}\right) $$

Substituting the given values:
$$ r = \sqrt{42^2 + 77^2} $$
$$ \theta = \arctan\left(\frac{77}{42}\right) $$

Once you compute these values, you will have the polar form $ (r, \theta) $ of the point (42, 77).