Problem 80 - Algebra and Trigonometry

Question

Convert each polar equation to a rectangular equation. Then determine the graph’s slope and y-intercept. $$ r \cos \left(\theta+\frac{\pi}{6}\right)=8 $$

Step-by-Step Solution

Answer

The slope is \( \sqrt{3} \) and the y-intercept is -16.

Step 1: Convert Polar to Rectangular Coordinates

Given the polar equation \(r \cos \left(\theta + \frac{\pi}{6}\right) = 8\). This can be converted to rectangular coordinates using the relationship \(r \cos(\theta) = x\). Hence, we can rewrite the equation in terms of x and y as \(x \cos\left(\frac{\pi}{6}\right) - y \sin\left(\frac{\pi}{6}\right) = 8\).

Step 2: Calculate cos and sin values

For \(\left(\frac{\pi}{6}\right)\), the cos value is \(\frac{\sqrt{3}}{2}\) and sin value is \(\frac{1}{2}\) . Thus, we replace these values in the earlier equation to get \(x\frac{\sqrt{3}}{2} - y\frac{1}{2} = 8\). This simplifies to \(\sqrt{3}x - y = 16\).

Step 3: Determine Slope and y-intercept

The equation is now in the form \(y = mx + c\), where m is the slope and c is the y-intercept. Rearranging our equation, we have \(y = \sqrt{3}x - 16\). Thus, the slope (m) is \(\sqrt{3}\) and the y-intercept (c) is -16.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, often known as Cartesian coordinates, consist of a system that denotes the position of points on a plane using two values: x and y. These coordinates are contrasted with polar coordinates, which use the distance from the origin (r) and the angle (θ) from a reference direction. To convert from polar to rectangular coordinates, certain relationships between the coordinates are used. Specifically:

• The x-coordinate is given by: \[x = r \cos(\theta)\]
• The y-coordinate is given by: \[y = r \sin(\theta)\]

In the provided exercise, the conversion involved using the given polar equation and adjusting it to fit into this rectangular format. This resulted in turning a polar equation into an equation that is linear in terms of x and y, specifically: \[\sqrt{3}x - y = 16\] after calculating cosine and sine values for \(\frac{\pi}{6}\). This conversion is crucial because it prepares the equation for further manipulation, such as finding the slope and y-intercept that are typically easier to identify in rectangular form.

Slope Calculation

The "slope" of a line in a Cartesian plane is a measure of its steepness and is a key feature in analyzing linear equations. The slope is generally represented by the letter m and can be found by rearranging a linear equation to the slope-intercept form: \[y = mx + c\]. In this form:

• "m" represents the slope of the line, and
• "c" is the y-intercept.

In our converted equation, obtained from the polar form:\[\sqrt{3}x - y = 16\] we rearranged it to get:\[y = \sqrt{3}x - 16\]. Here, it becomes quite clear that the slope, m, of the graph is \(\sqrt{3}\). This positive slope indicates that the line rises as it moves to the right on the graph. Understanding the slope is essential in predicting how the line behaves, especially when graphed against other lines or points of interest.

Y-intercept Determination

The y-intercept of a line is the point at which the line crosses the y-axis. This is a crucial aspect because it tells us where the line will start if you were drawing it from the bottom upwards on a graph. In the slope-intercept form \(y = mx + c\), the y-intercept is denoted by the c-value.Extracting the y-intercept from our equation \(y = \sqrt{3}x - 16\), we find that c is \(-16\). So, the line meets the y-axis at the point (0, -16). This negative value indicates that the intercept is below the origin. Knowing the y-intercept assists in sketching the graph quickly and accurately, particularly when you're matching the graph with real-world data or problems.