Problem 86 - College Trigonometry

Question

Convert the equation from polar coordinates into rectangular coordinates. $$ r=3 \sin (\theta) $$

Step-by-Step Solution

Answer

The rectangular form of the equation is \((y - \frac{3}{2})^2 + x^2 = \frac{9}{4}\).

Step 1: Understand the Relationship between Polar and Rectangular Coordinates

Recognize that polar coordinates \((r, \theta)\) can be converted into rectangular coordinates \((x, y)\) using the relationships: - \(x = r \cos(\theta)\)- \(y = r \sin(\theta)\)- \(r^2 = x^2 + y^2\)

Step 2: Identify Equation Components in Polar Form

The given polar equation is \(r = 3 \sin(\theta)\). Identify that this equation involves \(r\) and \(\sin(\theta)\).

Step 3: Express \(\sin(\theta)\) in Terms of \(y\) and \(r\)

Use the relationship \(y = r \sin(\theta)\) to express \(\sin(\theta)\) as \(\sin(\theta) = \frac{y}{r}\). Substitute this into the polar equation.

Step 4: Substitute and Simplify

Substitute the expression for \(\sin(\theta)\) into the given equation: \(r = 3 \cdot \frac{y}{r}\). Multiply both sides by \(r\) to remove the fraction: \(r^2 = 3y\).

Step 5: Use \(r^2 = x^2 + y^2\) to Convert into Rectangular Form

Substitute \(r^2 = x^2 + y^2\) into the equation from Step 4: \(x^2 + y^2 = 3y\). Rearrange to get the equation in a more standard form: \(x^2 + y^2 - 3y = 0\).

Step 6: Additional Simplification (Optional)

If desired, complete the square for the \(y\) terms to rewrite the equation \((y - \frac{3}{2})^2 - \frac{9}{4} + x^2 = 0\). Add \(\frac{9}{4}\) to both sides to simplify further: \((y - \frac{3}{2})^2 + x^2 = \frac{9}{4}\).

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points in a plane using a distance from a reference point and an angle from a reference direction. Unlike the traditional rectangular system based on x and y axes, polar coordinates express locations in terms of radial distance and angular relationships. This makes them very useful for circular and angular relationships in mathematics and physics.

• r: Represents the radial distance from the origin. It tells you how far the point is from the starting point.
• θ (theta): Represents the angle in radians from the positive x-axis. It shows the direction in which the point is located relative to the origin.

Understanding polar coordinates helps with simplifying problems involving circles and angles, making them a powerful tool in geometry and trigonometry.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, rely on a two-axis system to define points in the plane using horizontal and vertical distances. Named after the French mathematician René Descartes, this system lays the foundation for most of geometry and is widely used due to its straightforward representation of space.

• x: Horizontal axis coordinate. Indicates the position left or right of the origin.
• y: Vertical axis coordinate. Shows the position above or below the origin.

Rectangular coordinates provide an intuitive way to plot and analyze points and shapes on a graph. They are particularly beneficial when dealing with linear equations and when plotting functions.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for every value obtained within the applicable range. They are fundamental tools in mathematics, especially useful when transitioning between polar and rectangular coordinates.

• sin(θ): Determines the vertical component of a point in polar coordinates, expressed as \(sin(θ) = \frac{y}{r}\).
• cos(θ): Determines the horizontal component, noted as \(cos(θ) = \frac{x}{r}\).
• Pythagorean identity: Expresses the relationship between x and y in a circle, primarily \(x^2 + y^2 = r^2\).

These identities are crucial for converting between coordinate systems because they allow you to express polar descriptions in terms of x and y, which makes solving related problems much easier.

Coordinate Conversion

Coordinate conversion involves switching between different systems of loci representation: from polar (r, θ) to rectangular (x, y), or vice versa. This conversion is essential when the context or problem at hand naturally fits one representation over the other, allowing for simpler solutions and a deeper understanding of spatial relationships.To convert from polar to rectangular coordinates:

1. Use the formulas \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\) to drive the transition.
2. Identify \(\sin(\theta)\) and \(\cos(\theta)\) in terms of given values, facilitating the transition to x and y formats.

The reverse involves recognizing x and y in terms of r and θ, using the inverse trigonometric functions and maintaining the inherent relationships through Pythagorean identity. Understanding both methods significantly bolsters problem-solving capabilities when dealing with equations or systems involving circular and rotational elements.