Convert the Cartesian equation y=√3x to a polar equation.

Question

would it be...

a. r=√3θ
b. θ=√3r
c. θ=pi/3
d. r=√3

Answer 1

The easy way:  This is a line equation passing through origin with a slop √3,  and θ = Arctan(√3) = pi/3

   Therefore, answer is c, θ=pi/3. 

Another way:   y= rsinθ, x=  rcosθ and rewrite  equation to √3x - y =0

  Then, √3rcosθ - rsinθ = 0,  r(√3cosθ - sinθ )= 0,  r= 0 or √3cosθ - sinθ = 0

  The line passes through origin r=0 , but continuous. 

So, √3cosθ - sinθ = 0,  

   sinθ/cosθ = tanθ = √3,  and θ=pi/3

  

Answer 2

x = rcosθ and y = rsinθ, where 

 

So, y = √3 x can be rewritten as rsinθ = √3 rcosθ

 

                                               sinθ = √3 cosθ

 

Divide both sides by cosθ to get tanθ = √3

 

                                                  θ = tan^-1(√3)

 

                                                  θ = π/3

 

Answer 3

y=√3x

x=rcosθ

y=rsinθ

rsinθ=√3(rcosθ)

rsinθ/rcosθ=√3

sinθ/cosθ=√3

when θ=∏/3, cosθ=1/2

when θ=∏/3, sinθ=√3/2

sinθ/cosθ=(√3/2)/(1/2)=√3

c. θ=∏/3