Problem 3 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Question: Convert the Cartesian coordinate (x, y) to its corresponding polar coordinate (r, θ) using the derived equations. Answer: To convert a Cartesian coordinate (x, y) to its corresponding polar coordinate (r, θ), use the equations: \[r = \sqrt{x^2 + y^2}\] \[\theta = \arctan\frac{y}{x}\]

Step 1: Draw a Right Triangle

Draw a right triangle with the Cartesian point (x, y) as one of the vertices. The point (x, y) can also be represented as a vector with its origin at the origin of the coordinate system (0, 0) and its terminal point at (x, y). The polar coordinates (r, θ) can be represented by the length of this vector and the angle it forms with the positive x-axis.

Step 2: Use Pythagorean Theorem to Find 'r'

In the right triangle formed in Step 1, the length 'r' (the distance between the origin and the point (x, y)) can be found using the Pythagorean theorem. The relationship between x, y, and r can be represented as follows: \[r^2 = x^2 + y^2\] Therefore, we can find 'r' by taking the square root of both sides: \[r = \sqrt{x^2 + y^2}\]

Step 3: Use Trigonometric Ratios to Find 'θ'

In the right triangle formed in Step 1, we can use the trigonometric ratios to relate the angle θ with the coordinates (x, y). Specifically, we can use the tangent function, which is the ratio of the opposite side to the adjacent side in a right triangle: \[\tan\theta = \frac{y}{x}\] To find θ, we take the arctangent, or inverse tangent, of both sides: \[\theta = \arctan\frac{y}{x}\]

Step 4: Combine Equations for Polar Coordinates

We can now combine the equations derived in Steps 2 and 3 to express a point in Cartesian coordinates (x, y) in polar coordinates (r, θ): \[r = \sqrt{x^2 + y^2}\] \[\theta = \arctan\frac{y}{x}\] These equations can be used to convert any given Cartesian coordinate (x, y) to its corresponding polar coordinate (r, θ).