Problem 56 - Algebra and Trigonometry

Question

Polar coordinates of a point are given. Find the rectangular coordinates of each point. $$ \left(-3.1, \frac{91 \pi}{90}\right) $$

Step-by-Step Solution

Answer

(3.1, -0.1947)

Step 1: Understanding the Relationship Between Polar and Rectangular Coordinates

To convert polar coordinates \(r, \theta\) to rectangular coordinates \(x, y\), the formulas are \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). Here, \(r = -3.1\) and \(\theta = \frac{91 \pi}{90}\).

Step 2: Calculate the x-coordinate

Using the formula \(x = r \cos(\theta)\), substitute \(r = -3.1\) and \(\theta = \frac{91 \pi}{90}\). This gives: \[ x = -3.1 \cos\left( \frac{91 \pi}{90} \right) \approx -3.1 \cdot (-0.998) \approx 3.1 \]

Step 3: Calculate the y-coordinate

Using the formula \(y = r \sin(\theta)\), substitute \(r = -3.1\) and \(\theta = \frac{91 \pi}{90}\). This gives: \[ y = -3.1 \sin\left( \frac{91 \pi}{90} \right) \approx -3.1 \cdot 0.0628 \approx -0.1947 \]

Step 4: Combine the Results

The rectangular coordinates are obtained by combining the calculated values of \(x\) and \(y\). Hence, the rectangular coordinates are approximately \( (3.1, -0.1947) \).

Key Concepts

Polar Coordinates

Polar coordinates represent a point in the 2D plane using a distance and an angle. They are written as \( (r, \theta) \), where \( r \) is the radial distance from the origin, and \( \theta \) is the angle measured counterclockwise from the positive x-axis.
Here are some key points about polar coordinates:

• \( r \) can be positive or negative. A negative \( r \) value indicates the point is in the opposite direction from the angle \( \theta \).
• The angle \( \theta \) is typically given in radians, but can also be in degrees.

To better understand this, imagine you're standing at the origin (0,0). If someone tells you to walk 3 meters (that’s \( r \)) at an angle of 45 degrees (that’s \( \theta \)), those are your polar coordinates!

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, represent a point using its horizontal and vertical distances from the origin. They are written as \( (x, y) \). The coordinates tell you where to move along the x-axis (horizontal) and y-axis (vertical) to reach the point.
Here are key points about rectangular coordinates:

• \( x \) is the distance along the x-axis, which can be positive (right) or negative (left).
• \( y \) is the distance along the y-axis, which can be positive (up) or negative (down).

For example, if you have the point (4, 3), it means you go 4 units to the right along the x-axis and 3 units up along the y-axis.
This coordinate system is widely used because it easily aligns with equations, graphs, and other mathematical functions.

Coordinate Transformation

Coordinate transformation refers to the process of converting coordinates from one system to another. In this context, we're converting from polar coordinates to rectangular coordinates.
The formulas to transform from polar to rectangular coordinates are:
\[ x = r \cos(\theta) \] \[ y = r \sin(\theta) \] Here’s how you can apply these formulas:

• Identify \( r \) and \( \theta \) from the given polar coordinates.
• Substitute \( r \) and \( \theta \) into the formulas for \( x \) and \( y \).
• Calculate the resulting values to get the rectangular coordinates.

Let's take the given example: \( (-3.1, \frac{91 \pi}{90}) \). Here, \( r = -3.1 \) and \( \theta = \frac{91 \pi}{90} \). By plugging these into the transformation formulas, we get:
\[ x = -3.1 \cos(\frac{91 \pi}{90}) \approx 3.1 \] \[ y = -3.1 \sin(\frac{91 \pi}{90}) \approx -0.1947 \] Hence, the rectangular coordinates are approximately (3.1, -0.1947).
Practice more problems to get a better understanding and build confidence!