Problem 6 - Precalculus with Limits

Question

In Exercises 5-18, plot the point given in polar coordinate sand find two additional polar representations of the point, using \(-2\pi<\theta<2\pi\). \(\left(3, \dfrac{5\pi}{4}\right)\)

Step-by-Step Solution

Answer

Two additional polar representations of the point \((3, \frac{5\pi}{4})\) are \((3, \frac{13\pi}{4})\) and \((-3, \frac{9\pi}{4})\).

Step 1: Plotting the provided point

Begin by plotting the provided point \((3, \frac{5\pi}{4})\). This point has a radius of 3 units and is positioned at an angle \(\frac{5\pi}{4}\) radians, or 225 degrees, relative to the positive x-axis.

Step 2: Find the first additional representation

To find an additional representation of the same point but using a different angle, the general formula \(r, θ + 2nπ\) can be applied. Here, you can let \(n=1\), which gives the point as \((3, \frac{5\pi}{4} + 2π) = (3, \frac{13\pi}{4})\). This is an equivalent representation of the same point.

Step 3: Find the second additional representation

Another representation can be to use the negative radius with an angle shifted by \(\pi\) radians. This is based on the general formula \(-r, θ + (2n+1)π\). Let \(n=0\), which gives the point as \((-3, \frac{5\pi}{4} + π) = (-3, \frac{9\pi}{4})\). You would observe this to be an identical point plotted on the polar coordinate plane.

Key Concepts

Angle Conversion

When dealing with polar coordinates, angles can be represented in radians or degrees. The angle in polar coordinates tells you how far to rotate from the positive x-axis. For example, an angle of \( \frac{5\pi}{4} \) radians can also be expressed in degrees. To convert radians to degrees, multiply by \( \frac{180}{\pi} \). Thus, \( \frac{5\pi}{4} \times \frac{180}{\pi} = 225 \) degrees. Understanding this conversion is crucial as it allows you to visualize and interpret angles easily in either metric.

• **Radians**: Used more in calculus and when deriving trigonometric identities.
• **Degrees**: Often used in simple geometry and where intuitive understanding is needed.

Always remember, one full rotation around a circle is \( 2\pi \) radians or 360 degrees. Being comfortable with converting between these systems is essential for mastering polar coordinates.

Radius Representation

The radius in polar coordinates represents the distance from the origin to the point. It can be positive or negative, which affects how the point is plotted.

- **Positive Radius**: A positive radius, as in point \((3, \frac{5\pi}{4})\), means moving directly to the position along the given angle.- **Negative Radius**: Conversely, a negative radius like \((-3, \frac{9\pi}{4})\) means moving in exactly the opposite direction along the same line, effectively flipping the point's position.

Understanding that radius can be manipulated like this allows for finding alternative representations of the same point. By altering the angle appropriately, you ensure the point remains in the correct location while effectively changing how it is represented on the polar plane.

Polar to Cartesian Conversion

Converting polar coordinates to Cartesian coordinates involves translating a point from a polar plane to a rectangular grid. This is done using trigonomic relationships.

• The x-coordinate is found by multiplying the radius by the cosine of the angle: \( x = r \cos(\theta) \).
• The y-coordinate is calculated using the sine of the angle: \( y = r \sin(\theta) \).

For example, converting \((3, \frac{5\pi}{4})\) into Cartesian coordinates, we find:

• \( x = 3 \cos(\frac{5\pi}{4}) = -\frac{3\sqrt{2}}{2} \).
• \( y = 3 \sin(\frac{5\pi}{4}) = -\frac{3\sqrt{2}}{2} \).

Understanding this conversion process is important for problems where you need to switch between coordinate systems, like when graphing or solving physics problems.

Equivalent Angles

In polar coordinates, a single point can have multiple equivalent representations. This is due to the periodic nature of angles. Adding or subtracting \( 2\pi \) radians (or 360 degrees) results in the same position.

• For instance, the point \( (3, \frac{5\pi}{4}) \) can be equivalently represented as \( (3, \frac{13\pi}{4}) \), simply by adding \( 2\pi \).
• Negative representations are also possible. For \( (-3, \frac{9\pi}{4}) \), the negative radius is countered by adding \( \pi \) radians, changing the orientation but not the point.

These alternate representations illustrate how a single point in polar coordinates doesn't have just one fixed identity, allowing for flexibility in angle and radius representation.