-in-polar-coordinates.jpeg "(3 4) In Polar Coordinates")
Representing (3, 4) in Polar Coordinates
The point (3, 4) in the Cartesian coordinate system represents a point 3 units to the right of the origin and 4 units above it. To convert this to polar coordinates, we need to find the distance from the origin (the radius) and the angle the point makes with the positive x-axis.
Finding the Radius
The radius, denoted by 'r', is the distance from the origin to the point. We can find it using the Pythagorean theorem:
r = √(x² + y²) = √(3² + 4²) = √(9 + 16) = √25 = 5
Therefore, the radius is 5.
Finding the Angle
The angle, denoted by 'θ', is the angle made by the line connecting the origin to the point and the positive x-axis. We can find it using the arctangent function:
θ = arctan(y/x) = arctan(4/3) ≈ 53.13°
However, it's important to note that the arctangent function only gives us the angle in the first or fourth quadrant. Since our point (3, 4) is in the first quadrant, this angle is correct.
Representing in Polar Coordinates
Therefore, the polar coordinates of the point (3, 4) are (5, 53.13°). This means the point is 5 units away from the origin at an angle of approximately 53.13° with the positive x-axis.
Conclusion
Converting Cartesian coordinates to polar coordinates involves finding the distance from the origin (the radius) and the angle the point makes with the positive x-axis. This process allows us to represent the same point in a different coordinate system, offering a different perspective on its location.