Convert the given Cartesian coordinates to polar coordinates with n>0,0 ≤θ<2 πRemember to consid

Convert the given Cartesian coordinates to polar coordinates...

Key Concepts

Trigonometric Functions

Trigonometry is central to the process of converting between coordinate systems. Functions such as arctan, sine, and cosine help in calculating the angle and verifying the accuracy of the conversion through relationships like x = r cos(?) and y = r sin(?).

Polar Coordinates

This concept involves representing points in the plane using a radius and an angle, rather than x and y values. It is essential to understand how a point's position can be depicted by its distance from the origin and the direction relative to the positive x-axis.

Conversion Formulas

These are the mathematical relationships used to switch between Cartesian and polar coordinates. The primary formulas are r = sqrt(x² + y²) for the radial distance and ? = arctan(y/x) for the angle. Adjustments may be necessary to accurately reflect the point's position in its specific quadrant.

Quadrant Determination

When converting coordinates, it's crucial to determine which quadrant the point lies in based on the signs of its Cartesian coordinates. This affects the calculation of the angle, since the basic arctan function often returns angles that need to be adjusted to reflect the actual position of the point in the plane.

Transcript

00:01 Okay, this problem gives us a cartesian coordinate of negative 4 ,6, and it wants us to convert it to polar.

00:07 And so i wrote over here, right, i write the point saying we need to go to polar.

00:11 But then when we do this, i kind of wrote a little, like, review thing.

00:15 It's basically that triangle that we're making when we do it.

00:19 You go over x amount, up y amount, and that gives you a radius of r, right? in pythagorean's theorem gives us x squared plus y squared equals r squared, since it makes a right triangle with theta being that angle right there by the origin.

00:32 So i square rooted that to get r.

00:34 I'm going to be using this equation, right, because polar has an r and, whoops, that's not an r, r and then theta.

00:42 Okay, so i'm going to be using that to get r.

00:44 And then if you take the tangent of theta, that's your opposite over your adjacent or your y over x.

00:51 And then whenever you want the angle, use the arc or the inverse.

00:54 So i'm going to be using the inverse tangent to find the theta.

00:57 So those two equations are super important.

00:59 And so now i just plug it in, right? so r equals, who we go, r equals the square root of x squared, plus y squared, which is six squared.

01:11 And if you do that, that gives you the square root of 16 plus 36, which is the square root of 52.

01:19 Now, i don't know if you have to do this by hand or using a calculator.

01:22 So i'm just going to do it by hand.

01:23 And if you have a calculator, great, just type this in.

01:25 Okay, but 52, let's simplify that...

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