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Chapter 10: Problem 97

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

Short Answer

Expert verified
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 by step solution

01

Identify the Numerator and Denominator

Here the numerator is '4' and the denominator is '2 + cos \(\theta\)'. We need to divide both by 2.
02

Divide the Numerator by 2

Divide the numerator 4 by 2 to get 2.
03

Divide the Denominator by 2

Divide the denominator '2 + cos \(\theta\)' by 2 to get '1 + \(\frac{cos \(\theta\)}{2}\)'.
04

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}}\)'

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. You told me that an ellipse centered at the origin has vertices at \((-5,0)\) and \((5,0),\) so 1 was able to graph the ellipse.

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Identify the conic and write its equation in rectangular coordinates: \(r=\frac{1}{2-2 \cos \theta}\)

In each exercise, graph the equation in a rectangular coordinate system. $$\frac{x^{2}}{25}+\frac{y^{2}}{4}=1$$

Solve by eliminating variables: $$\left\\{\begin{aligned} x-6 y &=-22 \\ 2 x+4 y-3 z &=29 \\ 3 x-2 y+5 z &=-17 \end{aligned}\right.$$
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