How do you convert the rectangular coordinate #(-4.26,31.1)# into polar coordinates?

Question

Josiah Anderson · Accepted Answer

#(31.3,pi/2)# Changing to polar coordinates means we have to find #color(green)((r,theta))#. Knowing the relation between rectangular and polar coordinates that says: #color(blue)(x=rcostheta and y=rsintheta)# Given the rectangular coordinates: #x=-4.26 and y=31.3# #x^2+y^2=(-4.26)^2+(31.3)^2# #color(blue)((rcostheta )^2)+color(blue)((rsintheta)^2)=979.69# #r^2cos^2theta+r^2sin^2theta=979.69# #r^2(cos^2theta+sin^2theta)=979.69# Knowing the trigonometric identity that says: #color(red)(cos^2theta+sin^2theta=1)# We have: #r^2*color(red)1=979.69# #r=sqrt(979.69)# #color(green)(r=31.3)# Given: #color(blue)y=31.3# #color(blue)(rsintheta)=31.3# #color(green)31.3*sintheta31.3# #sintheta=31.3/31.3# #sintheta=1# #color(green)(theta=pi/2)# Therefore,the polar coordinates are #(color(green)(31.3,pi/2))#

Carter Anderson · Answer

undefined To convert the rectangular coordinates $(-4.26, 31.1)$ into polar coordinates, you can use the following formulas:

$$ r = \sqrt{x^2 + y^2} $$
$$ \theta = \arctan\left(\frac{y}{x}\right) $$

Where:
- $ r $ is the distance from the origin to the point.
- $ \theta $ is the angle between the positive x-axis and the line connecting the origin to the point.

Plugging the values $(-4.26, 31.1)$ into these formulas:

$$ r = \sqrt{(-4.26)^2 + (31.1)^2} $$
$$ r = \sqrt{18.1476 + 967.21} $$
$$ r = \sqrt{985.3576} $$
$$ r \approx 31.39 $$

$$ \theta = \arctan\left(\frac{31.1}{-4.26}\right) $$
$$ \theta = \arctan(-7.2976) $$
$$ \theta \approx -80.49^\circ $$

Therefore, the polar coordinates of $(-4.26, 31.1)$ are approximately $(31.39, -80.49^\circ)$.