Problem 322 - Calculus Volume 3

Question

True or False? Justify your answer with a proof or a counterexample. The rectangular coordinates of the point \(\left(4, \frac{5 \pi}{6}\right)\) are \((2 \sqrt{3},-2)\).

Step-by-Step Solution

Answer

False, the calculated coordinates don't match the given ones.

Step 1: Understanding Polar to Rectangular Conversion

To determine whether the given statement is true or false, we need to understand how to convert polar coordinates to rectangular coordinates. For a point given in polar coordinates \((r, \theta)\), the rectangular coordinates \((x, y)\) can be found using the formulas \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\).

Step 2: Calculating Rectangular Coordinates

Given the polar coordinates \((4, \frac{5\pi}{6})\), calculate the rectangular coordinates using the conversion formulas: - \(x = 4 \cos\left(\frac{5\pi}{6}\right)\)- \(y = 4 \sin\left(\frac{5\pi}{6}\right)\).Now calculate the values:\[\cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2}\]\[\sin\left(\frac{5\pi}{6}\right) = \frac{1}{2}\]Thus, \[x = 4 \left(-\frac{\sqrt{3}}{2}\right) = -2\sqrt{3}\]\[y = 4 \left(\frac{1}{2}\right) = 2\].

Step 3: Compare with Given Coordinates

We found the rectangular coordinates to be \((-2\sqrt{3}, 2)\). Now, compare this result with the given coordinates \((2 \sqrt{3}, -2)\) in the problem. They are not the same, as the x-coordinate and y-coordinate are the negatives of each other.

Step 4: Conclusion

Since the calculated rectangular coordinates \((-2\sqrt{3}, 2)\) do not match the proposed coordinates \((2\sqrt{3}, -2)\), the statement is false.

Key Concepts

Rectangular Coordinates

Rectangular coordinates are a way to represent a point in a two-dimensional plane using two values, often denoted as \((x, y)\). The first value \(x\) represents the horizontal distance from the origin, while the second value \(y\) represents the vertical distance from the origin.

• If \(x\) is positive, it means the point is to the right of the origin.
• If \(x\) is negative, it is to the left.
• Similarly, a positive \(y\) places the point above the origin, whereas a negative \(y\) places it below.

Rectangular coordinates are part of Cartesian coordinate systems, which form the backbone of coordinate geometry. Converting between different coordinate systems, such as polar and rectangular, allows us to understand geometric relationships and analyze graphs more thoroughly. When converting polar coordinates to rectangular coordinates, it is crucial to compute the horizontal and vertical components precisely.

Polar Coordinates

Polar coordinates provide an alternative way to represent points in a plane using a distance and an angle. In polar coordinates, a point is defined as \((r, \theta)\). Here, \(r\) is the distance from the point to the origin, and \(\theta\) is the angle between the positive x-axis and the line connecting the origin to the point.

• \(r\) is always non-negative.
• \(\theta\) is measured in radians, which are based on the radius of a circle.

Angles in polar coordinates can represent directions, making them useful in various fields like navigation and physics. Polar coordinates are particularly helpful when dealing with problems that involve circular or rotational symmetry. When converting polar coordinates to rectangular, trigonometric functions such as cosine and sine are used to find the respective \((x, y)\) components.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines algebra and geometry to describe geometric figures through equations and coordinates. It serves as a bridge between algebraic and geometric practices, allowing precise calculations and easier problem-solving.

• In coordinate geometry, points, lines, curves, and shapes are all expressed using algebraic equations.
• The Cartesian coordinate system is commonly employed.

Understanding how to move between different coordinate systems, like polar and rectangular systems, expands our ability to explore diverse mathematical scenarios. This dual-capability is invaluable in calculus, physics, and computer graphics, where plotting points and trajectories accurately is essential.

Trigonometric Conversion

Trigonometric conversion plays a crucial role in translating between polar and rectangular coordinates. The conversion relies heavily on the trigonometric functions \(\cos\) (cosine) and \(\sin\) (sine):

• To find the x-coordinate in rectangular form, use: \(x = r \cos(\theta)\).
• To find the y-coordinate, use: \(y = r \sin(\theta)\).

These formulas are derived from the relationship between trigonometric functions and the sides of a right triangle formed by dropping a perpendicular from the point to the x-axis. By leveraging these functions, every point in polar coordinates can be accurately projected into rectangular form, enabling seamless integration into Cartesian systems. Conversion mistakes can lead to incorrect positioning on the plane, emphasizing the need for precision in calculations and understanding each component's physical meaning.