Find the polar coordinates of the point (7, -7)?

Question

Please explain/show how you got the answer. Thanks :)

Answer 1

The coordinates (x , y) = (7, -7)

The polar coordinates (rcosθ , rsinθ)

r² = x² + y² = 7² + (-7)² = 98

Take square root to each side

r = √x² + y² = √98

tanθ = y / x

θ = arctan(y / x) = arctan(-7 / 7) = arctan(-1) = 3π/4,7π/4

Substitute r = √98 and θ = 3π/4,7π/4

The polar coordinates are (√98cos3π/4   , √98sin3π/4) and (√98cos7π/4   , √98sin7π/4).

 

Comments

The rectanglular coordinate point : (x, y)

The polar coordinate point : (r, θ) = [ √(x² + y²), tan^-1(y/x) ].

Since x = r cos(θ), y = r sin(θ) and tan(θ) = y/x.

r = √(x² + y²) = √[ (7)² + (- 7)² ] = √[49 + 49) = 7√2.

θ = tan^-1[ (- 7)/(7) ] = tan^-1(- 1) = - tan^-1(1) = - π/4.

The polar form of (7, - 7) is (r, θ) = (7√2, - π/4).