Problem 38 - Technical Mathematics with Calculus

Question

Write the rectangular coordinates of each point. $$\left(-12.0,-48.0^{\circ}\right)$$

Step-by-Step Solution

Answer

\((-12.0 \cos(-48.0^\circ), -12.0 \sin(-48.0^\circ))\)

Step 1: Understand Polar Coordinates

Polar coordinates describe a point in terms of an angle and a distance from the origin. They are given in the form \( (r, \theta) \), where \( r \) is the distance from the origin to the point and \( \theta \) is the angle measured in degrees or radians from the positive x-axis.

Step 2: Convert Angle to Radians

Check if the angle is in degrees or radians. In this case, \( -48.0^\circ \) is given in degrees. Convert it to radians using the conversion factor \( \frac{\pi}{180} \) since \( \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \) where \( \pi \) radians equals \( 180^\circ \).

Step 3: Compute Rectangular Coordinates

Use the polar to rectangular coordinate conversion formulas: \[ x = r \cos( \theta ) \] \[ y = r \sin( \theta ) \] Plug in the given \( r = -12.0 \) and the converted angle \( \theta \) in radians to find \( x \) and \( y \) coordinates.

Key Concepts

Polar Coordinates

Polar coordinates offer a different approach to describing the position of points in a plane compared to the more common Cartesian or rectangular coordinate system. While rectangular coordinates use a grid to specify a point's distance along the horizontal (x) and vertical (y) axes, polar coordinates specify a point's position based on its distance from a reference point (usually the origin) and the angle it makes with the positive x-axis.

Polar coordinates are denoted as \( (r, \theta) \) where \( r \) is the radial distance from the origin and \( \theta \) is the angle in standard position, usually measured in either degrees or radians. The angle originates from the positive x-axis, sweeping counter-clockwise for positive angles and clockwise for negative angles, which is an important aspect to consider during conversions.

Angle Conversion

When working with polar coordinates, converting angles between degrees and radians is a common necessity, especially since the trigonometric functions in rectangular coordinate conversions are often defined in terms of radians. The basic relationship between degrees and radians is that 360 degrees make a full circle, which is equivalent to \( 2\pi \) radians. Therefore, to convert from degrees to radians, you multiply by \( \frac{\pi}{180} \) and to convert from radians to degrees, you multiply by \( \frac{180}{\pi} \).

It's crucial to perform this conversion accurately as it directly impacts the results of trigonometric functions used to find rectangular coordinates. For instance, to convert the angle of \( -48.0^\circ \) into radians, you would calculate \( -48.0\times\frac{\pi}{180} \) which simplifies to \( -\frac{4\pi}{30} \) or approximately \( -0.8378 \) radians.

Rectangular Coordinates

Once you have polar coordinates and any necessary angle conversions complete, you can translate those polar coordinates into rectangular coordinates. This conversion uses the cosine and sine trigonometric functions with the given radius \( r \) and angle \( \theta \) to determine the x and y positions, respectively, in the Cartesian plane.

The formulas for this conversion are:

• \( x = r\cos(\theta) \) for the x-coordinate,
• \( y = r\sin(\theta) \) for the y-coordinate.

Applying these formulas requires you to take into account the angle in radians, ensuring that trigonometric calculations are accurate. For example, if we have \( -12.0 \) for \( r \) and \( -0.8378 \) radians for \( \theta \), we calculate the x-coordinate as \( -12.0\cos(-0.8378) \) and the y-coordinate as \( -12.0\sin(-0.8378) \) to find the rectangular coordinates of the point.