Problem 70 - College Trigonometry

Question

60\( through \)\\# 63\(. In Exercises 60 - 63 , you need to solve for … # Convert the equation from rectangular coordinates into polar coordinates. Solve for \)r\( in all but \)\\# 60\( through \)\\# 63\(. In Exercises 60 - 63 , you need to solve for \)\theta$. $$ x^{2}+y^{2}-2 y=0 $$

Step-by-Step Solution

Answer

In polar coordinates: \(r = 2 \sin \theta\).

Step 1: Express Known Relation Between Coordinates

Recall the conversion relations between rectangular coordinates \((x,y)\) and polar coordinates \((r, \theta)\): 1. \(x = r \cos \theta\)2. \(y = r \sin \theta\)3. \(x^2 + y^2 = r^2\).

Step 2: Substitute Known Polar Relation

Substitute the known relation \(x^2 + y^2 = r^2\) into the given equation:\[ r^2 - 2y = 0 \].

Step 3: Substitute Polar Expression for y

Substitute \(y = r \sin \theta\) into the equation:\[ r^2 - 2(r \sin \theta) = 0 \].

Step 4: Simplify the Equation

Reorganize the equation:\[ r^2 - 2r \sin \theta = 0 \].

Step 5: Factor Out Common Terms

Factor out \(r\) to solve the equation:\[ r(r - 2 \sin \theta) = 0 \].

Step 6: Analyze the Solutions

The factored equation \(r(r - 2 \sin \theta) = 0\) gives two solutions:1. \(r = 0\)2. \(r = 2 \sin \theta\) (but since \(r\) should generally not be zero unless trivial, use the second condition when \(r eq 0\)).

Step 7: Solve for θ

Since problem states particularly solve for \(\theta\), recognize that solution \(r = 2 \sin \theta\) requires analyzing possible values for \(\theta\). This condition represents a unit circle scaled and guided by halve the sine component of \(\theta\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates are a way of representing points on a plane using two numbers, typically \(x\) and \(y\). These coordinates form the basis of the Cartesian coordinate system, where \(x\) represents the horizontal axis, and \(y\) represents the vertical axis.

In our problem, the given equation \(x^2 + y^2 - 2y = 0\) is expressed in these rectangular coordinates. This equation defines a circle centered at the point \( (0, 1) \) with a radius of 1. This is determined by completing the square on \(y\), transforming the equation into \(x^2 + (y - 1)^2 = 1\).

Rectangular coordinates are particularly useful in problems where straightforward x, y directions are involved or when working with standard geometric shapes like lines and circles. Understanding them is the first step in transitioning to other coordinate systems such as polar coordinates.

Coordinate Conversion

Once we have our equation in rectangular form, we can transform it into polar coordinates. This process, known as coordinate conversion, involves utilizing specific formulas linking rectangular and polar coordinates.

The key formulas for conversion are:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)
• \(x^2 + y^2 = r^2\)

By substituting these into the rectangular equation, we can express it in terms of polar coordinates \(r\) and \(\theta\). For example, in our equation, using \(x^2 + y^2 = r^2\) gives \(r^2 = 2r \sin \theta\). Simplifying this, we factor out \(r\), leading to \(r(r - 2 \sin \theta) = 0\).

Coordinate conversion is essential in problems where the geometry of the problem fits naturally with circles or radial distances. It's a powerful tool in calculus and physics, offering an alternative way of viewing a problem.

Trigonometric Identities

Trigonometric identities play a crucial role in converting coordinate systems and solving equations in polar form. In our problem solution, we used the identity \(y = r \sin \theta\), which directly relates the rectangular y-coordinate to its polar counterpart.

Here are some valuable trigonometric relationships and identities often used in conversions and simplifications:

• \(\sin^2 \theta + \cos^2 \theta = 1\)
• \(1 + \tan^2 \theta = \sec^2 \theta\)
• \(1 - \sin^2 \theta = \cos^2 \theta\)

These identities help simplify complex expressions and solve for unknowns in polar equations. In our transformed equation \(r = 2 \sin \theta\), finding specific \(\theta\) values requires understanding and manipulating these identities.

Mastering these identities is important in tackling a plethora of mathematical problems, whether they're involving simple conversions or more complex trigonometric equations. They are especially important when terms like \(\sin \theta\) or \(\cos \theta\) appear in your equations and need simplification.