Problem 8 - Precalculus

Question

Plot the point given in polar coordinates and find two additional polar representations of the point, using \(-2 \pi < {\theta} < 2 \pi\). $$(-1,-3 \pi / 4)$$

Step-by-Step Solution

Answer

After plotting the point \(-1, -3 \pi / 4\), the additional polar representations of the point, considering the condition \(-2 \pi < θ < 2 \pi\), are \((-1, 1 \pi / 4)\) and \((-1, -15 \pi / 4)\).

Step 1: Plotting in Polar Coordinates

The point in polar coordinates is given as \((-1, -3 \pi / 4)\). To plot this point, counterclockwise direction is considered positive and clockwise negative. But the distance r from the origin cannot be negative. So, consider the angle as \(-3 \pi / 4 - \pi = -7 \pi / 4\). That is in the positive direction.

Step 2: Finding the Cartesian coordinates

Convert the polar coordinates to Cartesian coordinates using formulas \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). So, \(x = -1 \cos(-7 \pi / 4) = -\sqrt{2}/2\) and \(y = -1 \sin(-7 \pi / 4) = \sqrt{2}/2\)

Step 3: Find Additional Polar Representations

An additional polar representation can be found by adding or subtracting \(2 \pi\) to/from the existing \(θ\), staying within the boundary \(-2 \pi < θ < 2 \pi\). So, the two additional representations of the point are \((-1, 2 \pi - 7 \pi / 4) = (-1, 1 \pi / 4)\) and \((-1, -2 \pi - 7 \pi / 4) = (-1,-15 \pi / 4)\)

Key Concepts

Cartesian Coordinates

In mathematics, the Cartesian coordinate system is a foundational part of geometry. It is named after the French mathematician René Descartes, who revolutionized the concept of location in space using numbers.

Cartesian coordinates represent points in a plane using two numerical values: the x-coordinate and the y-coordinate.

• X-coordinate: This shows how far a point is along the horizontal axis or the x-axis.
• Y-coordinate: This indicates a point's location on the vertical axis, or y-axis.

Together, these coordinates provide a precise location in a 2D space, such as \( (x, y) \). This method is crucial for graphing equations and plotting points in a straightforward and visual manner.

In the given exercise, understanding the Cartesian system is essential after converting a point from polar to Cartesian coordinates. This conversion allows us to visualize the point on a familiar grid layout rather than on polar grids.

Coordinate Conversion

Coordinate conversion refers to changing a point from one form of coordinate representation to another. Switching between polar and Cartesian coordinates is a common type of conversion, chosen based on the context or specific needs of a problem.

**From Polar to Cartesian**: To convert a point given in polar coordinates to Cartesian coordinates, use the formulas: \( x = r \cos(\theta) \) for the x-coordinate and \( y = r \sin(\theta) \) for the y-coordinate. Here, \( r \) is the radial distance from the origin, and \( \theta \) is the angle from the positive x-axis.

**From Cartesian to Polar**: Conversely, you can convert by finding \( r \) using \( r = \sqrt{x^2 + y^2} \), and \( \theta \) using \( \theta = \tan^{-1}(y/x) \).

In the provided exercise, converting \( (-1, -3 \pi / 4) \) from polar to Cartesian results in coordinates \( (x, y) = \left(-\sqrt{2}/2, \sqrt{2}/2\right) \). This step is crucial for visual plotting on a Cartesian grid, providing insight into the position of the point in a recognizable format.

Angle Measure

Angles are a way to describe rotation from a specific reference point, usually the positive x-axis in mathematics. There are different methods to express angles, mainly in degrees and radians.

**Radians**: A radian is a mathematical unit of angular measure where the angle created is subtended at the center of a circle by an arc. One full circle is \( 2\pi \) radians. It's often used in more theoretical mathematics because it aligns closely with other mathematical principles, such as trigonometric functions.

• Positive angles move counterclockwise.
• Negative angles move clockwise.

In the exercise, the angle \( -3\pi/4 \) needed to be understood and adjusted within the allowable range of \( -2\pi < \theta < 2\pi \). The adjustment helps locate the angle in standard position without changing its overall position relative to the origin. Mastering angle measure is critical not only in plot conversion but also for broader applications in trigonometry and physics.