Converting Polar Coordinates to Cartesian Coordinates: A Step-by-Step Guide

Understanding how to convert polar coordinates to Cartesian coordinates is crucial in various mathematical and engineering applications. In this article, we will walk through the process of converting the polar coordinates (5, π - tan-1(4/3)) to Cartesian coordinates (x, y). This process involves using trigonometric identities and basic algebraic manipulation.

Introduction to Polar and Cartesian Coordinates

Polar coordinates are represented by a point's distance from the origin (radius, r) and an angle (theta, θ) with respect to the positive x-axis. Cartesian coordinates, on the other hand, are represented by a point's perpendicular distances from the x-axis (x-coordinate) and the y-axis (y-coordinate).

The conversion formulas from polar coordinates to Cartesian coordinates are as follows:

x r ? cos(θ) y r ? sin(θ)

Step-by-Step Conversion Process

Identify the given polar coordinates: r 5 θ π - tan-1(4/3)

Step 1: Calculate tan-1(4/3)

The value of tan-1(4/3) gives us an angle in the first quadrant, where the tangent of this angle is 4/3.

Step 2: Find cos(θ) and sin(θ)

Using the properties of trigonometric identities, we can express cos(θ) and sin(θ) in terms of the tangent:

cos(θ) cos(π - tan-1(4/3)) -cos(tan-1(4/3)) sin(θ) sin(π - tan-1(4/3)) sin(tan-1(4/3))

Step 3: Calculate sin(tan-1(4/3)) and cos(tan-1(4/3))

Using the triangle corresponding to the tangent value (opposite side 4, adjacent side 3):

Opposite side 4 Adjacent side 3 Hypotenuse √(42 32) √(16 9) √25 5

Therefore, we have:

sin(tan-1(4/3)) 4/5 cos(tan-1(4/3)) 3/5

Step 4: Substitute back to find cos(θ) and sin(θ)

Now substituting these values back:

cos(θ) -3/5 sin(θ) 4/5

Step 5: Calculate x and y

Now we can calculate x and y:

x r ? cos(θ) 5 ? (-3/5) -3 y r ? sin(θ) 5 ? (4/5) 4

Conclusion

The Cartesian coordinates (x, y) corresponding to the polar coordinates (5, π - tan-1(4/3)) are:

(-3, 4)

This step-by-step process demonstrates the mathematical reasoning and trigonometric identities required to convert polar coordinates to Cartesian coordinates. Understanding this conversion is essential in many real-world applications, such as navigation, physics, and engineering.