Question

Write the equations that are used to express a point with polar coordinates \((r, \theta)\) in Cartesian coordinates.

Short answer

Answer: The equations relating polar coordinates (r, θ) to Cartesian coordinates (x, y) are: x = r * cos(θ) y = r * sin(θ)

Step-by-step solution

Understand the relation between polar and Cartesian coordinates

Polar coordinates \((r, \theta)\) represent the position of a point in the plane by its distance \(r\) from the origin of the coordinate system and the angle \(\theta\) that the line connecting the origin and the point makes with the positive x-axis. On the other hand, Cartesian coordinates \((x, y)\) represent the position of a point in the plane by its distances from the x-axis (positive or negative) and from the y-axis (positive or negative).

Use trigonometry to relate polar and Cartesian coordinates

In order to find the relationship between polar and Cartesian coordinates, we can create a right triangle by connecting the point with both the x and y axes. Suppose we have a point with the polar coordinates \((r, \theta)\). We can form a right triangle with the point, the origin, and the intersection of the vertical line through the point and the x-axis. The length of the side of this triangle adjacent to angle \(\theta\) is given by the x-coordinate of the point, whereas the length of the side opposite to angle \(\theta\) is given by the y-coordinate of the point. Using the definitions of trigonometric functions, we can write the following equations: - cosine of angle \(\theta\), i.e. \(\cos{\theta}\), is equal to the ratio of the adjacent side to the hypotenuse, i.e. \(\frac{x}{r}\), so \(x = r \cos{\theta}\). - sine of angle \(\theta\), i.e. \(\sin{\theta}\), is equal to the ratio of the opposite side to the hypotenuse, i.e. \(\frac{y}{r}\), so \(y = r \sin{\theta}\).

Write the final equations

The desired Cartesian coordinates \((x, y)\) of a point given its polar coordinates \((r, \theta)\) are given by the following equations: \(x = r \cos{\theta}\) \(y = r \sin{\theta}\)