Problem 19 - Precalculus : Building Concepts and Connections

Question

Convert each of the given pairs of polar coordinates to a pair of rectangular coordinates. $$\left(3, \frac{\pi}{4}\right)$$

Step-by-Step Solution

Answer

The rectangular coordinates are \((2.1213, 2.1213)\).

Step 1: Identify the values of r and θ

Given the polar coordinates as \((3, \frac{\pi}{4})\), let \(r=3\) and \(\theta = \frac{\pi}{4}\). These are the values we will plug into our conversion formulas.

Step 2: Apply the conversion formulas

Now, apply the conversion formulas \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\) to convert the polar coordinates to rectangular coordinates. For \(x\), the calculation will be \(x = 3\cos(\frac{\pi}{4})\). For \(y\), the calculation will be \(y = 3\sin(\frac{\pi}{4})\).

Step 3: Calculate the results

By calculating the above expressions: \(x = 3\cos(\frac{\pi}{4}) = 3*0.7071 = 2.1213\) and \(y = 3\sin(\frac{\pi}{4}) = 3*0.7071 = 2.1213\) respectively.

Step 4: Write the final answer

Now, write the rectangular coordinate pair as \(x\) and \(y\) values respectively. Thus, the rectangular coordinates are \((2.1213, 2.1213)\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a way to describe a point in a two-dimensional plane using two values: the x-coordinate and the y-coordinate. These values represent distances along the horizontal and vertical axes, respectively. This system is named after the French mathematician René Descartes.

For a point \(x, y\), the x-coordinate shows how far the point is from the vertical axis, and the y-coordinate shows the distance from the horizontal axis. This clear-cut grid system is widely used in mathematics, engineering, and computer graphics to define the position of a point or to plot graphs.

To grasp rectangular coordinates, think of them as instructions to start from the origin (0,0) at the center of the plane and move:

• x units right if x is positive, or left if x is negative.
• y units up if y is positive, or down if y is negative.

These movements result in the precise location of a point on the plane.

Polar Coordinates

Polar coordinates provide an alternative method to describe a point in a plane using a radius and an angle. Instead of moving straight out along the x and y axes, polar coordinates use a distance from a central point (called the pole, similar to the origin in the rectangular system) and an angle from a reference direction (usually the positive x-axis).

The polar coordinate system can be particularly handy for situations involving circular motion or scenarios where symmetry around a central point is prevalent. A point in polar coordinates is denoted as \(r, \theta\), where:

• \(r\) is the radial distance from the pole (always non-negative).
• \(\theta\) is the angle in radians measured from the positive x-axis.

When converting from polar to rectangular, you use trigonometry to translate these curved paths into straight-line movements. This is crucial for applications in physics, engineering, and navigation.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles and sides of a right-angled triangle. They play a fundamental role in converting between polar and rectangular coordinates.

Key trigonometric functions include sine (\(\sin\)) and cosine (\(\cos\)), which are used to find horizontal and vertical components of an angle. Given a radius \(r\) and angle \(\theta\):

• \(x = r\cos(\theta)\) determines the horizontal x-coordinate.
• \(y = r\sin(\theta)\) determines the vertical y-coordinate.

By applying these functions, you translate the circular nature of polar coordinates into the linear format of rectangular coordinates. This conversion process is not only essential in theoretical mathematics but also applied disciplines like physics, where angles and distances often need precise calculations.

Understanding these trigonometric basics helps in translating angles into real-world distances or vice versa, thus forming the bridge between two coordinate systems.