Question

A point p is given in cartesian coordinates by p(−1, 1). Find polar coordinates (r, θ) of this point with r < 0 and 0 ≤ θ &lt; 2π.

Answer 1

The radius is the distance from (0, 0). It is found using the Pythagorean theorem.

r = √((-1)²+1²) = √2

The angle θ is found from the fact that r·sin(θ) = y = 1, and r·cos(θ) = x = -1. We take the ratio of these to get

... tan(θ) = 1/-1

... θ = arctan(-1) = 3π/4 . . . . . considering quadrant

Then the polar coordinates (r, θ) are (√2, 3π/4). You can multiply these by 1 = (-1, π) to get a negative value for r:

... (-1, 1) ⇒ (-√2, 7π/4)

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Multiplying numbers in polar coordinates multiplies their magnitudes and adds their angles.