Problem 17 - Calculus Early Transcendentals

Question

Express the following polar coordinates in Cartesian coordinates. \(\left(1,-\frac{\pi}{3}\right)\)

Step-by-Step Solution

Answer

Question: Convert the polar coordinates (1, -π/3) to Cartesian coordinates. Answer: The Cartesian coordinates are (1/2, -√3/2).

Step 1: Convert the polar coordinate to Cartesian coordinates

We are given the polar coordinate \((r, \theta) = \left(1,-\frac{\pi}{3}\right)\). We will plug these values into the conversion formulas to find the Cartesian coordinates \((x, y)\). \(x = r\cos\theta = 1\cos\left(-\frac{\pi}{3}\right)\) and \(y = r\sin\theta = 1\sin\left(-\frac{\pi}{3}\right)\)

Step 2: Evaluate the Trigonometric Functions

Evaluate the cosine and sine functions in the conversion formulas. \(x = \cos\left(-\frac{\pi}{3}\right)\) As cosine function is an even function, so \(\cos(-\theta)=\cos\theta\). Thus, \(x = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\) \(y = \sin\left(-\frac{\pi}{3}\right)=-\sin\left(\frac{\pi}{3}\right)=-\frac{\sqrt{3}}{2}\)

Step 3: Write the Cartesian coordinates

Now that we have found the values of \(x\) and \(y\), we can write the Cartesian coordinates. The Cartesian coordinates are \((x, y) = \left(\frac{1}{2},-\frac{\sqrt{3}}{2}\right)\).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, especially in the context of converting polar coordinates to cartesian coordinates. They relate the angles of a triangle to the lengths of its sides and have applications in various fields such as physics, engineering, and astronomy.

Cartesian Coordinate System

The Cartesian coordinate system is a two-dimensional coordinate system created by the French mathematician René Descartes. In this system, each point on a plane is determined by an ordered pair of numbers, known as coordinates. The first number (x-coordinate) measures the distance along the horizontal axis, while the second number (y-coordinate) measures the distance along the vertical axis. The Cartesian system is widely used in algebra and calculus for representing geometrical shapes, functions, and other mathematical expressions graphically.

Polar Coordinate System

The polar coordinate system is an alternative to the Cartesian coordinate system. It represents each point on a plane with two numbers: the radial distance from a fixed point (the pole, analogous to the origin in the Cartesian system) and an angle from a fixed direction (the polar axis, typically the line that represents the 0-degree angle). The radial distance is denoted as r, while the angle, usually measured in radians, is represented by θ. Polar coordinates are particularly useful for dealing with scenarios where phenomena are circular or radial in nature, such as in the orbits of celestial bodies or the vibration patterns of musical instruments.