What is the polar form of #( -6,36 )#?

1 Answer
May 1, 2018

#" "#
Cartesian Coordinates to Polar Form: #color(blue)((-6, 36) = (36, 99.4^@)#

Explanation:

#" "#
Given the Cartesian Form: #(-6, 36)#

Find the Polar Form:#color(blue)((r,theta)#

#color(green)("Step 1:"#

Let us examine some of the relevant formula in context:

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#color(green)("Step 2:"#

Plot the coordinate point #color(blue)((-6, 36)# on a Cartesian coordinate plane:

Indicate the known values, as appropriate:

enter image source here

#bar(OA)=6" Units"#

#bar(AB)=36" Units"#

Let #bar(OB)=r" Units"#

#/_OAB=90^@#

Let #/_AOB=alpha^@#

#color(green)("Step 3:"#

Use the formula: #color(red)(x^2 + y^2=r^2# to find #color(blue)(r#

Consider the following triangle with known values:

enter image source here

#r^2=6^2+36^2#

#rArr 36+1296#

#rArr 1332#

#r^2=1332#

Hence, #color(brown)(r=sqrt(1332)~~36.4966#

To find the value of #color(red)(theta)#:

#tan(theta)=36/6=6#

#theta= tan^-1(6)#

#theta ~~ 80.53767779^@#

#color(blue)("Important:"#

Since the angle #color(red)(theta# lines in Quadrant-II, we must subtract this angle from #color(red)180^@# to get the required angle #color(blue)(beta#.

#color(green)("Step 4:"#

enter image source here

#color(blue)(beta ~~ 180^@ - 80.53767779^@#

#rArr beta ~~ 99.46232221^@#

Hence, the required Polar Form:

#color(blue)((r, theta) = (36, 99.4^@)#

Hope it helps.