Problem 40 - Algebra and Trigonometry

Question

Polar coordinates of a point are given. Find the rectangular coordinates of each point. $$ (8.3,4.6) $$

Step-by-Step Solution

Answer

The rectangular coordinates of the point are approximately (-0.683, 8.266).

Step 1: Identify the given polar coordinates

In the given problem, the polar coordinates of the point are given. Identify the values of r (distance from the origin) and \( \theta \) (angle in radians). In this case, the polar coordinates are (8.3, 4.6), so \( r = 8.3 \) and \( \theta = 4.6 \).

Step 2: Apply conversion formulas

Next, apply the conversion formulas which are \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \) to get the rectangular coordinates.

Step 3: Substitute r and \( \theta \) into conversion formulas

Substitute the values of r and \( \theta \) from Step 1 into the formulas. This gives \( x = 8.3 \cos(4.6)\) and \( y = 8.3 \sin(4.6)\).

Step 4: Calculate x and y

Now, find the values of x and y using a calculator. It's important to calculate the cosine and sine values in radian mode because the angle \( \theta \) is given in radians. In this case, the values are \( x = -0.683 \) and \( y = 8.266 \).

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points in a plane using a circle-based system. Rather than using a grid, like in the rectangular (Cartesian) coordinate system, polar coordinates describe positions based on an angle and a distance from a fixed point, known as the pole. The polar system includes:

• r: The distance from the pole (origin) to the point.
• \(\theta\): The angle formed between the polar axis (usually the positive x-axis in rectangular coordinates) and the line connecting the pole to the point.

Representing a point in polar coordinates looks like \((r, \theta)\). For example, if a point has coordinates \((8.3, 4.6)\), it means the point is 8.3 units away from the origin and the angle \(\theta\) is 4.6 radians. Unlike degrees, radians express angles in terms of π and are often used in mathematical formulas. This system is particularly useful in scenarios involving circular or spiral patterns, like polar graphs or in physics.

Rectangular Coordinates

Rectangular coordinates, or Cartesian coordinates, are a way of locating points on a two-dimensional plane using two fixed perpendicular lines, known as the x-axis and y-axis. Each point on the plane is described by a pair of coordinates:

• x-coordinate: This represents the horizontal distance from the origin.
• y-coordinate: This represents the vertical distance from the origin.

Every point is written in the form \((x, y)\). The origin, where the two axes intersect, is the point \((0, 0)\). If a point is located at \((x, y) = (-0.683, 8.266)\), it is approximately 0.683 units to the left of the y-axis and 8.266 units above the x-axis. Rectangular coordinates are widely used in many fields, from mathematics and engineering to everyday applications like mapping and navigation. They provide a straightforward way to calculate distances between points and are essential in geometric graphing.

Trigonometric Conversion Formulas

To convert between polar and rectangular coordinates, we use trigonometric functions, specifically sine and cosine. These functions relate the angle \(\theta\) and the distance \(r\) to our rectangular coordinates

• x-coordinate: Calculated using the formula \(x = r \cos(\theta)\)
• y-coordinate: Calculated using the formula \(y = r \sin(\theta)\)

In our example, given \(r = 8.3\) and \(\theta = 4.6\) radians, the rectangular coordinates are found by substituting into these formulas:

• \(x = 8.3 \cos(4.6)\)
• \(y = 8.3 \sin(4.6)\)

Calculating these gives us \((x, y) = (-0.683, 8.266)\). When performing these calculations, it's crucial to use radians as the angle measure if \(\theta\) is given in radians. Trigonometric conversions provide a bridge between the two systems, allowing you to move seamlessly between polar and rectangular graphs.