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Chapter 6: Problem 80

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

Short Answer

Expert verified
The rectangular form of the equation is \(y = x\sqrt{3} - 16\), with a slope of \(\sqrt{3}\) and a y-intercept of -16.

Step by step solution

01

Conversion to Rectangular Form

Start by substituting \(r \cos(\theta)\) as x in the given equation \(r \cos \left(\theta+\frac{\pi}{6}\right)=8\). This leads to \(x \cos\left(\frac{\pi}{6}\right) - y \sin\left(\frac{\pi}{6}\right) = 8\).
02

Simplify Equation

Next, simplify the equation by calculating the values of \(\cos\left(\frac{\pi}{6}\right)\) and \(\sin\left(\frac{\pi}{6}\right)\) as \(\frac{\sqrt{3}}{2}\) and \(\frac{1}{2}\) respectively. This leads to \(x \frac{\sqrt{3}}{2} - y \frac{1}{2} = 8\). Multiplying the whole equation by 2, we get: \(x\sqrt{3} - y = 16\).
03

Find Slope and y-Intercept

Finally, determine the slope and y-intercept of the obtained equation. The slope can be found by isolating y, and the y-intercept is the constant term of the equation. By rearranging, the equation is found to be \(y = x\sqrt{3} - 16\). thus the slope is \(\sqrt{3}\) and the y-intercept is -16.

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