Problem 34 - Precalculus Mathematics for Calculus

Question

Find the rectangular coordinates for the point whose polar coordinates are given. $$(\sqrt{3},-5 \pi / 3)$$

Step-by-Step Solution

Answer

The rectangular coordinates are \(\left(\frac{\sqrt{3}}{2}, -\frac{3}{2}\right)\).

Step 1: Understanding Polar Coordinates

The given polar coordinates are \((r, \theta) = (\sqrt{3}, -\frac{5\pi}{3})\). Here, \(r\) represents the distance from the origin to the point, and \(\theta\) represents the angle measured from the positive x-axis in the counter-clockwise direction.

Step 2: Converting to Rectangular Coordinates Formula

To convert polar coordinates \((r, \theta)\) to rectangular coordinates \((x, y)\), we use the formulas:\[ x = r \cos \theta \]\[ y = r \sin \theta \]These will give us the \(x\) and \(y\) components of the rectangular (Cartesian) coordinates.

Step 3: Substitute Values

Substitute \(r = \sqrt{3}\) and \(\theta = -\frac{5\pi}{3}\) into the formulas:\[ x = \sqrt{3} \cos \left(-\frac{5\pi}{3}\right) \]\[ y = \sqrt{3} \sin \left(-\frac{5\pi}{3}\right) \]

Step 4: Evaluate Cosine and Sine

Recall that the angle \(-\frac{5\pi}{3}\) is coterminal with \(-\frac{5\pi}{3} + 2\pi = \frac{\pi}{3}\), so we use the trigonometric values at \(\frac{\pi}{3}\):\[ \cos \left(\frac{\pi}{3}\right) = \frac{1}{2} \]\[ \sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]

Step 5: Compute x and y

Plug the values from the previous step into the equations for \(x\) and \(y\):\[ x = \sqrt{3} \times \frac{1}{2} = \frac{\sqrt{3}}{2} \]\[ y = \sqrt{3} \times \frac{\sqrt{3}}{2} = \frac{3}{2} \]

Step 6: Write the Rectangular Coordinates

The rectangular coordinates \((x, y)\) of the point are \(\left(\frac{\sqrt{3}}{2}, -\frac{3}{2}\right)\).

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points in a plane using a distance and an angle. The distance, denoted as \(r\), is the straight-line distance from the origin (or the center of the coordinate system) to the point. This is similar to a radius in a circle. The angle, \(\theta\), is how much the line from the origin to the point revolves around the origin. This angle is measured from the positive x-axis going counterclockwise. Here are some key points about polar coordinates:

• The angle \(\theta\) can be positive or negative. Positive angles are counterclockwise, while negative angles are clockwise.
• A point could be represented by different pairs of \((r, \theta)\) because angles that differ by full rotations (multiples of \(2\pi\)) represent the same direction.
• The coordinate system is very useful in scenarios involving circular and rotational motion.

Understanding polar coordinates is essential when dealing with problems involving circles and rotations, as they can simplify complex relationships between points.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are probably the most familiar coordinate system. They indicate a point's position using two numbers, \(x\) and \(y\). These two values describe a location in reference to two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Some important aspects of rectangular coordinates include:

• The coordinate \(x\) shows how far left or right a point is from the y-axis.
• The coordinate \(y\) reveals how far up or down a point is from the x-axis.
• The system is well-suited for describing points in a grid-like pattern, making it great for most everyday applications.

Understanding rectangular coordinates is key, as they are the foundation for most mathematical and scientific applications involving space and position.

Trigonometric Functions

Trigonometric functions play a crucial role in converting between polar and rectangular coordinates. These functions relate the angles of a triangle to the lengths of its sides and come in handy for situations like circular motion or as in our case, coordinate conversion. Key trigonometric functions include cosine and sine:

• Cosine (\(\cos\)): Relates the adjacent side of a right triangle to its hypotenuse and is used to determine the x-coordinate in conversions.
• Sine (\(\sin\)): Correlates the opposite side to the hypotenuse and is useful for finding the y-coordinate.

When converting from polar to rectangular coordinates, the formulas \(x = r \cos \theta\) and \(y = r \sin \theta\) allow you to switch between the two systems by exploiting the triangular relationships within a "unit circle", a circle with a radius of one. Understanding these functions not only eases coordinate conversions but also opens up possibilities in solving various mathematical problems involving angles and distances.