#" "#
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:
#color(green)("Step 2:"#
Plot the coordinate point #color(blue)((-6, 36)# on a Cartesian coordinate plane:
Indicate the known values, as appropriate:
#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:
#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:"#
#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.
