Problem 97 - Algebra and Trigonometry

Question

Rewrite \(r=\frac{4}{2+\cos \theta}\) by dividing the numerator and the denominator by 2.

Step-by-Step Solution

Answer

After dividing numerator and denominator by 2, the equation \(r=\frac{4}{2+\cos \theta}\) simplifies to \(r=\frac{2}{1 + \frac{cos \theta}{2}}\)

Step 1: Identify the Numerator and Denominator

Here the numerator is '4' and the denominator is '2 + cos \(\theta\)'. We need to divide both by 2.

Step 2: Divide the Numerator by 2

Divide the numerator 4 by 2 to get 2.

Step 3: Divide the Denominator by 2

Divide the denominator '2 + cos \(\theta\)' by 2 to get '1 + \(\frac{cos \(\theta\)}{2}\)'.

Step 4: Rewrite the equation

Replace the numerator and the denominator in the original fraction with the values obtained in steps 2 and 3. This gives the rearranged equation as 'r = \(\frac{2}{1 + \(\frac{cos \(\theta\)}{2}}\)'

Key Concepts

Understanding Trigonometric Functions

Trigonometric functions are fundamental in mathematics and are especially useful in dealing with angles and circles. The most common trigonometric functions include sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). In polar coordinates, these functions help describe the position of a point as an angle and a distance from a fixed point.

• **Cosine Function**: When you encounter \(\cos \theta\), it tells you about the x-coordinate of a point on a unit circle at angle \(\theta\).
• **Unit Circle**: A fundamental tool where the radius is 1. Often used to visualize and understand trigonometric functions.
• **Polar Coordinates**: Consist of a radius and an angle, usually represented as \((r, \theta)\).

Understand that trigonometric functions are cyclical and repeat every 360 degrees or \(2\pi\) radians. This repetition is crucial when solving problems that involve these functions, like those found in polar coordinates.

Simplifying Fractions in Equations

Fractions are a way to represent parts of a whole, and they consist of a numerator (top part) and a denominator (bottom part). When working with equations like polar coordinate transformations, having fractions makes the equation easier to manage and understand.

• **Numerator**: It is the part of the fraction that tells how many parts we have.
• **Denominator**: It tells the total number of equal parts the whole is divided into.
• **Manipulating Fractions**: You can simplify fractions by dividing both the numerator and the denominator by the same non-zero number.

This process helps to retain the value of the fraction, but in a simpler and more useful form. Simplifying fractions helps in making calculations easier and often reveals simplifications or truths about the problem.

Numerator and Denominator Simplification

In mathematics, simplifying the numerator and denominator is a crucial skill that often leads to solving many kinds of equations effectively. It's a straightforward process but it requires attention to detail to maintain the equality of the equation.

• **Step-by-step simplification**: Identify what each part of the fraction represents, as seen in the equation \(r = \frac{4}{2 + \cos \theta}\). Here, 4 is the numerator, and \(2 + \cos \theta\) is the denominator.
• **Dividing numerator**: Take the numerator 4 and divide it by 2 to simplify to 2.
• **Dividing denominator**: Apply the same division to each term within the denominator \(2 + \cos \theta\). Doing so, you get \(1 + \frac{\cos \theta}{2}\).

This careful handling keeps the fraction's value unchanged and often clearer to work with. Simplifying both parts ensures the equation is easier to interpret and solve, as seen when rewriting polar coordinate equations.