Problem 76 - 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$. $$ 4 x^{2}+4\left(y-\frac{1}{2}\right)^{2}=1 $$

Step-by-Step Solution

Answer

The polar coordinate for \(\theta\) is \(\pi/2\).

Step 1: Recall concepts of polar coordinates

Polar coordinates are represented as \((r, \theta)\). In these, \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). Substitute these into the given equation to convert from rectangular coordinates.

Step 2: Substitute expressions for x and y

Substitute \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\) in the given equation \[4x^{2}+4(y-\frac{1}{2})^{2}=1\] to get: \[4(r \cos(\theta))^{2} + 4 \left(r \sin(\theta) - \frac{1}{2}\right)^{2} = 1\]

Step 3: Simplify the equation

Expand the squared terms: \[4(r^2 \cos^2(\theta)) + 4 \left(r^2 \sin^2(\theta) - r \sin(\theta) + \frac{1}{4}\right) = 1\] Combine like terms: \[4r^2 (\cos^2(\theta) + \sin^2(\theta)) - 4r \sin(\theta) + 1 = 1\] Recall \(\cos^2(\theta) + \sin^2(\theta) = 1\), so the equation simplifies to: \[4r^2 - 4r \sin(\theta) + 1 = 1\]

Step 4: Solve for \(\theta\)

Simplify further by subtracting 1 from both sides: \[4r^2 - 4r \sin(\theta) = 0\] Factor out the \(4r\): \[4r(r - \sin(\theta)) = 0\]Since \(r = 0\) does not provide information about \(\theta\), solve for \(r - \sin(\theta) = 0\): \[r = \sin(\theta)\]. Extract \(\theta\) using arcsin function:\[\theta = \arcsin\left(\frac{r}{r}\right) \rightarrow \theta = \pi/2\] (since sine function equals one at \(\pi/2\)).

Key Concepts

Rectangular Coordinates

In a two-dimensional space, rectangular coordinates are used to determine the position of a point using two numbers, typically noted as \(x, y\). These coordinates are based on a system of perpendicular axes: one runs horizontally (the x-axis), and the other runs vertically (the y-axis). Any point within this space can be represented by its horizontal displacement (x-coordinate) and vertical displacement (y-coordinate) from a fixed reference point known as the origin, located at (0, 0). Rectangular coordinates are popular and commonly used in various applications such as plotting graphs, solving algebraic equations, and working with vector representations. To better understand why we convert from rectangular to polar coordinates, remember that some equations, like circles or spirals, are simpler to express using angles and distances, which is what polar coordinates offer.

Conversion Equations

The conversion from rectangular coordinates to polar coordinates involves a straightforward interchange using known formulas. Polar coordinates are given in the form \(r, \theta\), where \(r\) is the distance from the origin to the point and \(\theta\) is the angle from the positive x-axis to the line connecting the origin to the point.

• To convert x-coordinates: \(x = r \cos(\theta)\)
• To convert y-coordinates: \(y = r \sin(\theta)\)

These equations are vital because they allow for the transformation of points between systems, enabling simpler calculations in polar coordinates, especially for certain types of curves. During this conversion process, substitution is a key step. For example, replacing \(x\) and \(y\) in an equation with the equivalent polar expressions allows the transformation necessary to solve for either \(r\) or \(\theta\), depending on the requirement of the problem. In many situations, this conversion can significantly simplify the equation, making it easier to analyze geometrical properties or relationships.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values within their domains. They play a crucial role in simplifying expressions, especially when converting between rectangular and polar coordinates.One of the most fundamental trigonometric identities used in these conversions is:

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

This identity asserts that no matter the value of \(\theta\), the sum of the squares of the cosine and sine of \(\theta\) is always 1. This identity simplifies expressions involving both sine and cosine, especially when squared terms appear in the conversion process.By applying this identity, you can often reduce complex expressions significantly. In practice, this simplification helps in isolating terms to solve for variables, such as \(\theta\) in the polar form, further elucidating relationships between various trigonometric components.