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Given the Cartesian Form: (-6, 36)(−6,36)
Find the Polar Form:color(blue)((r,theta)(r,θ)
color(green)("Step 1:"Step 1:
Let us examine some of the relevant formula in context:
![enter image source here]()
color(green)("Step 2:"Step 2:
Plot the coordinate point color(blue)((-6, 36)(−6,36) on a Cartesian coordinate plane:
Indicate the known values, as appropriate:
![enter image source here]()
bar(OA)=6" Units"¯¯¯¯¯¯OA=6 Units
bar(AB)=36" Units"¯¯¯¯¯¯AB=36 Units
Let bar(OB)=r" Units"¯¯¯¯¯¯OB=r Units
/_OAB=90^@∠OAB=90∘
Let /_AOB=alpha^@∠AOB=α∘
color(green)("Step 3:"Step 3:
Use the formula: color(red)(x^2 + y^2=r^2x2+y2=r2 to find color(blue)(rr
Consider the following triangle with known values:
![enter image source here]()
r^2=6^2+36^2r2=62+362
rArr 36+1296⇒36+1296
rArr 1332⇒1332
r^2=1332r2=1332
Hence, color(brown)(r=sqrt(1332)~~36.4966r=√1332≈36.4966
To find the value of color(red)(theta)θ:
tan(theta)=36/6=6tan(θ)=366=6
theta= tan^-1(6)θ=tan−1(6)
theta ~~ 80.53767779^@θ≈80.53767779∘
color(blue)("Important:"Important:
Since the angle color(red)(thetaθ lines in Quadrant-II, we must subtract this angle from color(red)180^@180∘ to get the required angle color(blue)(betaβ.
color(green)("Step 4:"Step 4:
![enter image source here]()
color(blue)(beta ~~ 180^@ - 80.53767779^@β≈180∘−80.53767779∘
rArr beta ~~ 99.46232221^@⇒β≈99.46232221∘
Hence, the required Polar Form:
color(blue)((r, theta) = (36, 99.4^@)(r,θ)=(36,99.4∘)
Hope it helps.
