Which equation is the polar equivalent to the equation y=−3√3x? θ=π6 θ=π3 θ=2π3 θ=5π6

Answer 1

Answer:

Rearrange equation y = -3√3 x , y/x = -3√3 /1, ( x, y) = ( 1, -3√3)

The radius r is the hypotenuse of the right angle triangle r = √(x²)+(y²) = √( 1²+ (-3√3)² = √ 1+ 9(3) = √28 = 2√7 , and θ = tan^-1( y/x) = tan^-1( -3√3 / 1) = atan -3√3 or θ = atan( 1/-3√3) + 2πn = -atan3√3 + 2πn where n = number of 2π rotations

In the interval ( 2π, 4π) , θ = - atan( 3√3) + 2π(1) = - atan( 3√3) + 2π

or θ = - atan( 3√3) + 2π(2) = - atan( 3√3) + 4π

( r, θ ) = ( 2√7 , -atan 3√3 + 2π) , ( 2√7, 4.9 rad)

or if r is rotated counterclockwise 180°, = ( -2√7, -atan 3√3 + π) = ( -2√7, 1.76 rad)

in degrees θ = 4.9 rad ( 180 °/ π rad) = 280°, ( r, θ) = ( 2√7, 280º)

in degrees θ = 1.76 rad( 180°/ π rad) = 100º, ( r, θ) = ( -2√7, 100º)