Problem 27 - Precalculus with Limits

Question

In Exercises 19-28, a point in polar coordinates is given. Convert the point to rectangular coordinates. \(\left(-2.5, 1.1\right)\)

Step-by-Step Solution

Answer

The rectangular coordinates of the point are (-1.4, -2.27).

Step 1: Identify the given polar coordinates

The polar coordinates of the point are given as \((-2.5, 1.1)\), where -2.5 is the radius (r) and 1.1 is the angle in radians (theta).

Step 2: Convert the polar coordinates into rectangular form

Convert the polar coordinates into rectangular coordinates using the following formulas:\n \(x = r*\cos(\theta)\) and \(y = r*\sin(\theta)\). Plugging in our values, calculate \(x = -2.5*\cos(1.1)\) and \(y = -2.5*\sin(1.1)\).

Step 3: Carry out the calculation

After performing the calculation, we find that \(x \approx -1.4\) and \(y \approx -2.27\).

Key Concepts

Polar Coordinates

Polar coordinates define the position of a point in terms of a distance and an angle. This system is especially useful in situations where circular or rotational paths are central to the problem. The two values in polar coordinates are

• Radius (\(r\)): Represents the distance from the origin to the point.
• Angle (\(\theta\)): Shows the counterclockwise angle from the positive x-axis to the line connecting the origin with the point.

In the case of the given exercise, the point \((-2.5, 1.1)\) illustrates a radius of -2.5 and an angle of 1.1 radians. Negative radii reflect that the point is located in the opposite direction of the angle from the origin. To find where a point lies with polar coordinates, always consider both the direction and the length of the radius.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, use two perpendicular lines or axes to define the position of a point. These axes—normally denoted as the x-axis and the y-axis—form the traditional grid structure many are familiar with. In this system, the coordinates \((x, y)\) indicate horizontal and vertical distances from the origin \((0, 0)\).To convert a point from polar to rectangular coordinates, we use two key formulas:

• \(x = r\cos(\theta)\)
• \(y = r\sin(\theta)\)

For the point \((-2.5, 1.1)\), substituting the given values into the formulas, we obtain \(x \approx -1.4\) and \(y \approx -2.27\). These calculations represent how far along each of the axes the point is located.

Trigonometric Functions

Trigonometric functions were crucial in the conversion from polar to rectangular coordinates. They help in relating angles to side ratios in right triangles, which in turn aids in our calculations for coordinate transformation.For this purpose, the primary functions involved are:

• \(\cos(\theta)\): The cosine function relates the adjacent side of an angle in a right triangle to the hypotenuse.
• \(\sin(\theta)\): The sine function relates the opposite side of an angle in a right triangle to the hypotenuse.

The angle 1.1 radians, when used with the trigonometric functions, allows converting the radial length to the appropriate horizontal and vertical components. This way, we effectively change our perspective from a circular viewpoint to a grid-like structure suited to rectangular coordinates.