Question

In Exercises \(1-4,\) convert the point from cylindrical coordinates to rectangular coordinates. $$ (6,-\pi / 4,2) $$

Short answer

The rectangular coordinates of the given point in cylindrical coordinates (6, -\(\pi / 4\), 2) are \( (3\sqrt{2}, -3\sqrt{2}, 2)\).

Step-by-step solution

Identify the Cylindrical Coordinates

The given point is (6, -\(\pi / 4\), 2). Here, r = 6, theta = -\(\pi / 4\), and z = 2 are the cylindrical coordinates of the point.

Apply the Conversion Formulas

To convert these into rectangular coordinates, the formulas x = rcos(θ), y = rsin(θ), and z = z are employed. These formulas result from the basic properties of the unit circle and the definition of cosine and sine.

Compute the Coordinates

Substituting the values into the formulas we get, \(x = 6 * cos(-\pi / 4) = 6 * \sqrt{2} / 2 = 3\sqrt{2}\), \(y = 6 * sin(-\pi / 4) = 6 * -\sqrt{2} / 2 = -3\sqrt{2}\), and z = 2. Hence the rectangular coordinates are \( (3\sqrt{2}, -3\sqrt{2}, 2)\).

Key concepts

Coordinate Conversion

Understanding how to convert between different types of coordinate systems is an essential skill in higher mathematics, particularly when dealing with complex shapes and spaces. In this exercise, cylindrical coordinates are converted to rectangular coordinates, which can often simplify the solving of problems involving three-dimensional space.

Cylindrical coordinates, given as \( (r, \theta, z) \), are a set of three numbers that specify the position of a point in three-dimensional space. The value of \( r \) represents the radial distance from the origin, \( \theta \) is the angle in the xy-plane from the positive x-axis, and \( z \) provides the height above the xy-plane.

To convert these to rectangular coordinates \( (x, y, z) \) where \( x \) and \( y \) specify the horizontal position and \( z \) remains the same, indicating vertical position, the following formulas are applied:

Trigonometry in Calculus

The step of transition between different coordinate systems involves trigonometric functions, which form the basis of much of calculus, especially when working with shapes and motions in space. Trigonometry in calculus often comes into play when analyzing the properties of triangles and their relation to circular functions.

In converting cylindrical to rectangular coordinates, the formulas \( x = r\cos(\theta) \) and \( y = r\sin(\theta) \) are used. The angle \( \theta \) determines the direction of the point from the origin, while the radius \( r \) determines the distance from the origin. The cosine and sine functions provide the means to break this polar representation into x and y components, reflecting the horizontal and vertical displacements respectively from the origin in the x- and y-axes.

For instance, the cosine of angle \( \theta \) determines the x-coordinate by projecting the radial distance onto the x-axis, while the sine of \( \theta \) determines the y-coordinate by projecting the same radial distance onto the y-axis.

Unit Circle

The unit circle is a fundamental concept in trigonometry, represented by a circle of radius 1 centered at the origin of the coordinate plane. It serves as a visual aid for understanding angles and lengths in trigonometry. When using the unit circle, angles are measured from the positive x-axis, with counter-clockwise movement signifying positive angles and clockwise negative ones.

In this exercise, point’s angle is at \( -\pi / 4 \) radians, indicating a negative 45-degree rotation from the positive x-axis. The coordinates of the points on the unit circle corresponding to \( \cos(\theta) \) and \( \sin(\theta) \) give the x and y components for any radius \( r \) by multiplying these by \( r \). To find the coordinates for the given problem, utilize the values on the unit circle where the angle \( -\pi / 4 \) corresponds to the coordinates \( (\sqrt{2}/2, -\sqrt{2}/2) \) on the unit circle. Multiplying these by the radius \( r = 6 \) will acquire the rectangular coordinates for the provided cylindrical point.