Problem 80 - Algebra and Trigonometry

Question

The letters \(x\) and \(y\) represent rectangular coordinates. Write each equation using polar coordinates \((r, \theta) .\) $$ r=\sin \theta+1 $$

Step-by-Step Solution

Answer

y = r^2 - r

Step 1: Know the relationship between coordinates

To convert from rectangular coordinates \((x, y)\) to polar coordinates \((r, \theta)\), use the following equations: \(x = r\cos\theta\), \(y = r\sin\theta\), and \(r = \sqrt{x^2 + y^2}\).

Step 2: Analyze the given polar equation

The given equation is \(r = \sin\theta + 1\). Notice that the equation is already in terms of polar coordinates, where \(r\) (the radius) is expressed as a function of \(\theta\) (the angle).

Step 3: Rewrite the equation using known polar relationships

We know that \(r = \sqrt{x^2 + y^2}\) and \(\sin\theta = \frac{y}{r}\). Since the equation is given in polar form, let’s express \(\sin\theta\) in terms of \(r\) and \(y\). Substitute \(\sin\theta = \frac{y}{r}\) into the given equation to get \(r = \frac{y}{r} + 1\).

Step 4: Simplify the equation

Multiply both sides by \r\ to clear the fraction: \(r^2 = y + r\). Rearrange the equation to make \(y\) the subject: \(y = r^2 - r\).

Key Concepts

rectangular coordinates

Rectangular coordinates, often called Cartesian coordinates, use the \(x\) and \(y\) axes to locate points in a plane. These coordinates are typically written as \((x, y)\). In a rectangular coordinate system, \(x\) represents the horizontal distance from the origin (0,0), while \(y\) represents the vertical distance. This system is useful for plotting linear equations and is commonly used in algebra and calculus.
For example, consider the point (3, 4). This means moving 3 units along the x-axis and 4 units up the y-axis.

• The origin is where the x-axis and y-axis intersect: (0, 0).
• Positive x values are to the right, negative x values are to the left.
• Positive y values are upwards, negative y values are downwards.

coordinate conversion

Coordinate conversion is the process of changing from one coordinate system to another. In this context, it often refers to converting from rectangular (Cartesian) coordinates to polar coordinates, and vice versa. Understanding this conversion is crucial in trigonometry and calculus because some problems are easier to solve in polar form.

• To convert from rectangular to polar coordinates, use the formulas: \( r = \sqrt{x^2 + y^2} \) and \( \theta = \arctan\left(\frac{y}{x}\right) \).
• To convert from polar to rectangular coordinates, use: \ x = r\cos\theta \ and \ y = r\sin\theta \.

For example, if you have the rectangular coordinates \( (3, 4) \), you can convert them to polar coordinates as follows:

• Calculate \( r = \sqrt{3^2 + 4^2} = 5 \).
• Calculate \(\theta = \arctan\left(\frac{4}{3}\right)\) (roughly 53.13 degrees or 0.93 radians).

Therefore, the polar coordinates are approximately \( (5, 0.93) \).

equation transformation

Equation transformation involves rewriting an equation in a different coordinate system to make the problem easier to solve. In this exercise, we transform an equation given in polar coordinates into rectangular coordinates.
The given polar equation is: \( r = \sin\theta + 1 \).
Follow these steps to transform it:
1) Recognize the known relationships: \( x = r\cos\theta \, y = r\sin\theta \, r = \sqrt{x^2 + y^2} \).
2) Identify \( \sin\theta \) in terms of \( y \) and \( r \): \( \sin\theta = \frac{y}{r} \. \)
3) Substitute this into the given equation: \( r = \frac{y}{r} + 1 \. \)
4) Clear the fraction by multiplying both sides by \( r \): \( r^2 = y + r \. \)
5) Rearrange to make \( y \) the subject: \( y = r^2 - r \. \)
Through these transformations, we find that the polar equation represents a quadratic relationship between \( y \) and \( r \), which may simplify graphing or solving the equation in certain contexts.