Question

Indicate the direction in which the spiral winds outward as \(\theta\) increases, where \(\theta>0 .\) Let \(a=1\) and \(a=-1\) Hyperbolic spiral: \(r=a / \theta\)

Short answer

Answer: For \(a=1\), the hyperbolic spiral winds inward towards the origin as \(\theta\) increases. For \(a=-1\), the hyperbolic spiral winds outward away from the origin, but in the negative direction, as \(\theta\) increases.

Step-by-step solution

Understand the hyperbolic spiral equation

The given equation for the hyperbolic spiral is \(r = \frac{a}{\theta}\), where \(r\) is the radial distance from the origin and \(\theta\) is the angle in radians.

Calculate values of \(r\) for different values of \(\theta\) when \(a=1\)

As \(\theta\) increases from \(0\) to \(2\pi\), we calculate the values of \(r\) for \(a=1\) using the formula \(r = \frac{1}{\theta}\). Here are a few example points (rounded): - For \(\theta=0.1\), \(r=\frac{1}{0.1}=10\) - For \(\theta=1\), \(r=\frac{1}{1}=1\) - For \(\theta=2\), \(r=\frac{1}{2}=0.5\) - For \(\theta=3\), \(r=\frac{1}{3}=0.33\)

Determine the direction of the spiral when \(a=1\)

As we can see from the calculated values in Step 2, as \(\theta\) increases, the radial distance \(r\) decreases, and the spiral winds inward towards the origin.

Calculate values of \(r\) for different values of \(\theta\) when \(a=-1\)

As \(\theta\) increases from \(0\) to \(2\pi\), we calculate the values of \(r\) for \(a=-1\) using the formula \(r = \frac{-1}{\theta}\). Here are a few example points (rounded): - For \(\theta=0.1\), \(r=\frac{-1}{0.1}=-10\) - For \(\theta=1\), \(r=\frac{-1}{1}=-1\) - For \(\theta=2\), \(r=\frac{-1}{2}=-0.5\) - For \(\theta=3\), \(r=\frac{-1}{3}=-0.33\)

Determine the direction of the spiral when \(a=-1\)

As we can see from the calculated values in Step 4, as \(\theta\) increases, the radial distance \(r\) increases in the negative direction (the opposite side of the origin in polar coordinates), and the spiral winds outward away from the origin, but in the negative direction. In conclusion, the hyperbolic spiral winds inward towards the origin as \(\theta\) increases when \(a=1\), and it winds outward away from the origin (in the negative direction) as \(\theta\) increases when \(a=-1\).

Key concepts

Hyperbolic Spiral

In mathematics, the hyperbolic spiral is a fascinating curve specified in polar coordinates and described by the equation \(r = \frac{a}{\theta}\). In this context, \(r\) represents the radial distance from the origin, and \(\theta\) is the angle measured in radians. Here, \(a\) can be any real number, and its sign significantly affects the spiral's orientation.

When \(a = 1\), as \(\theta\) increases, the value of \(r\) decreases, showing that the spiral winds towards the origin. Conversely, if \(a = -1\), the radial distances also decrease as \(\theta\) increases, but in a negative direction, causing it to wind outward, away from the origin.

The distinction in the direction arises because of the sign of \(a\).

• If \(a\) is positive, the spiral converges inward.
• If \(a\) is negative, the spiral diverges outward.

This unique characteristic of hyperbolic spirals makes them an intriguing subject of study within the field of polar coordinates.

Radial Distance

Radial distance, denoted as \(r\), is a crucial component in any polar coordinate system. It defines the distance from the origin point (center) to a particular point on the curve. Unlike the Cartesian coordinate system, which employs \(x\) and \(y\) to locate points, polar coordinates utilize \(r\) and \(\theta\).

Understanding radial distance is fundamental in solving polar equations like the hyperbolic spiral \(r = \frac{a}{\theta}\). Here's why it's important:

• \(r\) informs us how far a point is from the origin.
• Changes in \(r\) based on \(\theta\) help determine the curve's shape and direction.

For the hyperbolic spiral, the value of \(r\) decreases with an increasing \(\theta\) if \(a\) is positive, signaling a winding towards the origin. For negative \(a\), \(r\) grows negatively relative to the angle, indicating expansion away from the origin.

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent are central to understanding angles and their relationships in polar coordinates. Although not directly used in defining the hyperbolic spiral, these functions can aid in manipulating polar equations and understanding more complex scenarios.

Polar coordinates express points based on angles and radial distances; hence, trigonometric functions are frequently utilized:

• \(\cos(\theta)\) can be used to find the x-coordinate of a point in a polar system.
• \(\sin(\theta)\) determines the y-coordinate, linking polar and Cartesian coordinates.

These functions also play a key role in converting between different coordinate systems and tackling higher-level applications involving rotations and transformations. Grasping these basics aids in expanding understanding beyond simple polar equations, opening doors to advanced mathematical explorations.