Problem 15 - Precalculus

Question

Plot the point given in polar coordinates and find two additional polar representations of the point, using \(-2 \pi < {\theta} < 2 \pi\). $$(\sqrt{2}, 2.36)$$

Step-by-Step Solution

Answer

The given point \((\sqrt{2}, 2.36)\) translates to point \((-1, 1)\) in Cartesian coordinates. Two additional polar representations for the same point are \((\sqrt{2}, 8.5)\) and \((\sqrt{2}, -3.92)\).

Step 1: Interpreting polar coordinates

The given polar coordinates are \((r, \theta) = (\sqrt{2}, 2.36)\). In polar coordinates, \(r\) represents distance from the origin and \(\theta\) is the angle formed with the positive x-axis. Therefore, from the origin, go \(\sqrt{2}\) units far and at an angle of \(2.36\) radian (approximately \(135^\circ\)). This point in Cartesian coordinates is \((-1, 1)\).

Step 2: Finding two additional polar representations

Two additional polar representations of the point can be obtained by adding or subtracting multiples of \(2\pi\) (full circle's radian measure) to/from the angle \(\theta\). Let's add \(2\pi\) to \(2.36\) for the first representation. We obtain \(2.36 + 2\pi \approx 8.5\). So, the first additional representation is \(\sqrt{2}, 8.5\). For the second representation, subtract \(2\pi\) from \(2.36\). We obtain \(2.36 - 2\pi \approx -3.92\). Therefore, the second additional representation is \(\sqrt{2}, -3.92\).

Step 3: Plotting the points

Three points to plot on the polar grid are \((\sqrt{2}, 2.36), (\sqrt{2}, 8.5), (\sqrt{2}, -3.92)\). All three points lie on the line that is \(\sqrt{2}\) units away from the origin, but differ in angle which indicates their location around the circle.

Key Concepts

Understanding Radians

Radians are a way to measure angles that's closely related to the radius of a circle. Instead of measuring angles in degrees, we use the distance traveled along the edge of a circle.
If you imagine a circle with a radius of 1, a full circle is about 6.28 units around, which is known as 2π radians.

Unlike degrees, which divide a circle into 360 parts, radians are an actual length around the circle's circumference. This helps in calculations related to circular motion or when converting back and forth between different mathematical representations.

Discovering Cartesian Coordinates

Polar coordinates mark a point by its distance and direction from the origin. On the other hand, Cartesian coordinates use a familiar grid system with an x and y-axis.

• Polar to Cartesian: The formula to convert polar \(r, \theta\) to Cartesian (x, y) is:
• \( x = r \cdot \cos(\theta) \)
• \( y = r \cdot \sin(\theta) \)

For this problem, the polar point \(\sqrt{2}, 2.36\) converts to Cartesian coordinates as \((-1, 1)\). This directional shift is crucial when transitioning between systems, ensuring proper placement in the grid.

Mastering Angle Conversion

Converting angles between systems is vital. In the polar system, this often means working with radians. To better understand, remember:

• 1 complete circle = 2π radians = 360 degrees.
• Common conversions: π radians = 180 degrees, π/2 radians = 90 degrees.

Adding or subtracting 2π helps find equivalent angles within specified ranges. For example, adding 2π to 2.36 radians gives you another valid angle while still representing the same direction, just past the standard circle range.

Exploring Multiple Representations

In polar coordinates, you can represent the same point in various ways by adjusting the angle. Given the angle restriction of \(-2\pi < {\theta} < 2\pi\), you explore other forms by modifying \({\theta}\).

• Add 2π to \({\theta}\) for a new circle rotation.
• Subtract 2π for the opposite direction's equivalent.

This ensures multiple polar representations relate back to the same Cartesian point. For example, the given point \(\sqrt{2}, 2.36\) can also be represented by angles \{8.5, -3.92\}. Each representation mirrors the same point, simply viewed from a different rotational perspective.