Problem 91 - Algebra and Trigonometry

Question

Explain how to convert a point from polar to rectangular coordinates. Provide an example with your explanation.

Step-by-Step Solution

Answer

The point in polar coordinates \(P(r,\theta) = P(3,\pi/4)\) is converted to rectangular coordinates as \(P(x,y) = P(3/\sqrt{2},3/\sqrt{2})\).

Step 1: Understand Polar and Rectangular Coordinate Systems

In a polar coordinate system, each point in a plane is determined by the distance from a reference point (usually the origin) and an angle from a reference direction (usually the x-axis). In rectangular (or Cartesian) coordinates, each point is determined by the horizontal (x) and vertical (y) distances from the origin.

Step 2: Learn the Conversion Formulas

To convert from polar to rectangular coordinates, use the formulas \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\), where \(r\) is the distance from the origin to the point, \(\theta\) is the angle from the x-axis to the line connecting the origin and the point, \(x\) is the horizontal distance from the origin, and \(y\) is the vertical distance from the origin.

Step 3: Apply the Conversion Formulas

Assume a point \(P\) in polar coordinates is \(P(r,\theta) = P(3, \pi/4)\). Applying the conversion formulas gives \(x = 3 \cos(\pi/4) = 3/\sqrt{2}\), and \(y = 3 \sin(\pi/4) = 3/\sqrt{2}\). So, the point \(P\) in rectangular coordinates is \(P(x,y) = P(3/\sqrt{2}, 3/\sqrt{2})\).