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The rectangular coordinates of the point P are (4, 3). What are the polar coordinates of this point?
Graph the point P and draw the triangle:
We need to find r and θ.
We use the Pythagorean theorem to find the hypotenuse:
To find θ we use the definition of the tangent function.
The polar coordinates of P are
(r, θ) = (5, 0.644).
The polar coordinates of the point Q are . What are the rectangular coordinates of Q?
It's always to wise to start by graphing the point and drawing a triangle:
We need to find x and y. To find y we use
To find x we use
The rectangular coordinates of Q are
If the point in question is on one of the axes, there's no need to draw a triangle because we can look and see what's going on.
What are the polar coordinates of the point with rectangular coordinates (5, 0)?
The point lies on the x-axis:
Since the point is on the positive side of the x-axis, we can take θ = 0. Since the point is a distance of 5 from the origin, r must be 5. The polar coordinates are
(r, θ) = (5, 0). They're exactly the same as the rectangular coordinates!
Translate the rectangular coordinates (-4, -4) into polar coordinates.
This point is in the third quadrant:
When we translate into polar coordinates, r should be positive and θ should be between π and .
Here's the useful triangle:
We find the magnitude r by finding the hypotenuse of the triangle:
Then we find the direction θ by finding the reference angle α and adding π:
The polar representation of the point is
This is in the correct quadrant, so it's a perfectly reasonable answer.
Translate the polar coordinates into rectangular coordinates.
This point is in the fourth quadrant, which means the rectangular coordinates should have x > 0 and y < 0.
We know that
We do mean θ here, not some reference angle.
y = r sin θ.
x = r cos θ.
To find x and y, we plug the values and into these equations.