-to-polar-form.jpeg "(-1 1) To Polar Form")
Converting Rectangular Coordinates (-1, 1) to Polar Form
In mathematics, especially when dealing with complex numbers and trigonometry, it's often useful to represent points in different coordinate systems. One common conversion is from rectangular coordinates (x, y) to polar coordinates (r, θ).
This article will guide you through the process of converting the point (-1, 1) from rectangular to polar form.
Understanding the Concepts
- Rectangular Coordinates (x, y): This system uses two perpendicular axes, the x-axis (horizontal) and the y-axis (vertical), to locate a point.
- Polar Coordinates (r, θ): This system uses a distance (r) from the origin and an angle (θ) measured counterclockwise from the positive x-axis to locate a point.
Steps to Convert (-1, 1) to Polar Form
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Calculate the Distance (r): The distance r is the length of the line segment connecting the origin (0, 0) to the point (-1, 1). We can find it using the Pythagorean theorem:
r = √(x² + y²) = √((-1)² + (1)²) = √2 -
Calculate the Angle (θ): We can use the arctangent function (tan⁻¹) to find the angle θ. Since the point (-1, 1) lies in the second quadrant, we need to add 180 degrees (or π radians) to the result to get the correct angle:
θ = tan⁻¹(y/x) + π = tan⁻¹(1/-1) + π = -π/4 + π = 3π/4
Result
Therefore, the polar form of the point (-1, 1) is (√2, 3π/4).
Note: You can express the angle in degrees (135°) or radians (3π/4). Remember to always choose the angle that corresponds to the correct quadrant where the point lies.