Problem 23 - Precalculus with Calculus Previews

Question

Find the rectangular coordinates for each point with the given polar coordinates. $$ (4, \pi / 8) $$

Step-by-Step Solution

Answer

The rectangular coordinates are approximately (3.6956, 1.5308).

Step 1: Understand the Problem

We need to convert the given polar coordinates \((r, \theta)\) where \(r = 4\) and \(\theta = \frac{\pi}{8}\) into rectangular coordinates \((x, y)\).

Step 2: Recall the Conversion Formulas

To convert from polar coordinates \((r, \theta)\) to rectangular coordinates \((x, y)\), we use the formulas: \[ x = r \cos(\theta) \] \[ y = r \sin(\theta) \]

Step 3: Calculate the x-coordinate

Substitute \(r = 4\) and \(\theta = \frac{\pi}{8}\) into the formula for \(x\):\[ x = 4 \cos\left(\frac{\pi}{8}\right) \]Calculate \(\cos\left(\frac{\pi}{8}\right)\) using trigonometry tables or a calculator to find:\[ x \approx 4 \times 0.9239 \approx 3.6956 \]

Step 4: Calculate the y-coordinate

Substitute \(r = 4\) and \(\theta = \frac{\pi}{8}\) into the formula for \(y\):\[ y = 4 \sin\left(\frac{\pi}{8}\right) \]Calculate \(\sin\left(\frac{\pi}{8}\right)\) using trigonometry tables or a calculator to find:\[ y \approx 4 \times 0.3827 \approx 1.5308 \]

Step 5: Write the Final Answer

Now that we have calculated both \(x\) and \(y\), we can write the rectangular coordinates: The rectangular coordinates are approximately \((3.6956, 1.5308)\).

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that studies relationships in triangles. It's all about angles and distances, and it plays a crucial role in coordinate transformation. At the heart of trigonometry are functions like sine, cosine, and tangent. These functions help us relate the angles of a triangle to the lengths of its sides.

• Sine (\(\sin\theta\)): Opposite side over hypotenuse.
• Cosine (\(\cos\theta\)): Adjacent side over hypotenuse.
• Tangent (\(\tan\theta\)): Opposite side over adjacent side.

For transformations, especially from polar to rectangular coordinates, cosine and sine are the stars of the show. By using these trigonometric functions, we can project points defined by an angle and distance from the origin onto the x and y axes.

Polar Coordinates

Polar coordinates describe a point in a plane using a distance from a reference point and an angle from a reference direction. In essence, they give you a vantage point from which you can draw a circle (radius) and measure an angle. This is different from the Cartesian system, which uses straightforward x and y distances along perpendicular axes.

• Radius (\(r\)): The straight-line distance from the origin to the point.
• Angle (\(\theta\)): The angle formed with the positive x-axis, often measured in radians.

Polar coordinates are particularly useful in contexts where circular or rotational symmetry is involved, such as in engineering and physics problems. They make it easy to describe concentric circles and spirals.

Rectangular Coordinates

Rectangular coordinates, often referred to as Cartesian coordinates, denote a point in the plane in terms of two numbers: its horizontal and vertical distances from the origin. This system breaks down space into a grid using an x-axis and a y-axis:

• x-coordinate: Represents how far the point is horizontally from the y-axis.
• y-coordinate: Represents how far the point is vertically from the x-axis.

This method is intuitive for navigation on a map, designing layouts, or graphing functions, as it directly aligns with how we visualize space in two dimensions. In the context of conversions, these coordinates are often the end goal of transforming polar coordinates to a system that's easier to plot or use in algebraic calculations.

Coordinate Transformation

Coordinate transformation involves converting the representation of a point from one coordinate system to another. It can seem daunting, but it's fundamentally about changing perspectives:

• Polar to Rectangular: Uses the formulas \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \) to express a point in terms of x and y.
• Rectangular to Polar: Converts back using \( r = \sqrt{x^2 + y^2} \) and \( \theta = \tan^{-1}(y/x) \).

The process helps us leverage the strengths of both systems. Polar coordinates can simplify problems involving rotations or circular paths, while rectangular coordinates make it easier to analyze straight-line relationships. Each system offers unique advantages depending on the context of the problem.