Problem 34 - Precalculus

Question

A point in polar coordinates is given. Convert the point to rectangular coordinates. $$(-2,5.76)$$

Step-by-Step Solution

Answer

The rectangular coordinates of the point (-2, 5.76) in polar coordinates are approximately (\(-2 \times \cos(5.76)\), \(-2 \times \sin(5.76)\)).

Step 1: Calculate the x-coordinate

To calculate the x-coordinate in rectangular system, you use the formula \(x = r \times \cos(\theta)\). Therefore, in this case, the x-coordinate is given by \(x = -2 \times \cos(5.76)\).

Step 2: Calculate the y-coordinate

To find the y-coordinate in rectangular coordinate system, the formula \(y = r \times \sin(\theta)\) is used. Hence, the y-coordinate is given by \(y = -2 \times \sin(5.76)\).

Step 3: Evaluate the results

Using an appropriate calculator that can calculate sines and cosines, calculate the x and y coordinates from Step 1 and Step 2.

Key Concepts

Polar Coordinates

Polar coordinates represent a point in a two-dimensional plane using a distance and an angle. Instead of using x and y like in rectangular coordinates, polar coordinates use \(r\) and \(\theta\).

• The distance \(r\) is the radial distance from the origin (center point) to the point.
• The angle \(\theta\) is measured from the positive x-axis, moving counter-clockwise.

Polar coordinates are written in the form \((r, \theta)\). This system is particularly useful in scenarios where circular or rotational symmetry exists. For instance, in the exercise given, \((-2, 5.76)\) refers to a point that is 2 units away from the origin but in the opposite direction because of the negative radius, and the angle 5.76 radians from the positive x-axis.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, specify a point using horizontal and vertical distances. These distances are labeled as x and y coordinates.

• The x-coordinate tells you how far left or right the point is from the y-axis.
• The y-coordinate shows how far up or down the point is from the x-axis.

This system is familiar because it resembles the grid system used in many graphing scenarios. In the context of converting polar to rectangular coordinates, you'll calculate these x and y values using trigonometric functions. For instance, converting \((-2, 5.76)\) involves applying the conversion formulas: \[ x = r \times \cos(\theta) \] \[ y = r \times \sin(\theta) \] These formulas take the radial distance and angle and translate them into horizontal and vertical distances that are used in rectangular coordinates.

Trigonometric Functions

Trigonometric functions are fundamental in converting polar coordinates to rectangular coordinates. The primary functions used in this conversion are cosine and sine.

• Cosine (cos): This function relates the angle \(\theta\) to the x-coordinate, calculated by the formula \(x = r \times \cos(\theta)\).
• Sine (sin): This function relates the angle \(\theta\) to the y-coordinate, calculated by the formula \(y = r \times \sin(\theta)\).

These functions essentially project the polar coordinates onto the x and y axes: - The cosine function determines the horizontal distance (x-component).- The sine function determines the vertical distance (y-component).Calculators or software that can evaluate trigonometric functions are very helpful when working with angles that are not standard angles (like the \(5.76\) in our exercise). Using these functions, you can move seamlessly between polar and rectangular coordinate systems.