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Chapter 10: Problem 99

Rotated Curve Verify that if the curve whose polar equation is \(r=f(\theta)\) is rotated about the pole through an angle \(\phi,\) then an equation for the rotated curve is \(r=f(\theta-\phi)\)

Short Answer

Expert verified
The equation of the curve after rotating it by an angle \(\phi\) about the pole is \(r = f(\theta - \phi)\)

Step by step solution

01

Analysis of the situation

A curve in polar coordinates is defined by the equation \(r = f(\theta)\), where \(r\) represents the distance from the origin (or the pole) and \(\theta\) is the angle created with the positive x-axis. When the curve rotates an angle \(\phi\) about the pole, the angle \(\theta\) originally with the x-axis will then be \(\theta - \phi\) with the x-axis.
02

Understanding the transformation

After the rotation, every angle \(\theta\) will now be \(\theta - \phi\). Hence the relationship \(r = f(\theta)\) would also transform according to the change in the angle.
03

Finding the equation of the rotated curve

Replacing \(\theta\) by \(\theta - \phi\) in our original equation, we get \(r = f(\theta - \phi)\). This equation represents how the function changes its angles due to the rotation, implying that this is the equation of the curve after it has been rotated by an angle \(\phi\) about the pole.

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Most popular questions from this chapter

Finding the Arc Length of a Polar Curve In Exercises \(57-62,\) use a graphing utility to graph the polar equation over the given interval. Use the integration capabilities of the graphing utility to approximate the length of the curve accurate to two decimal places. $$ \begin{array}{ll}{\text { Polar Equation }} & {\text { Interval }} \\\ {r=e^{\theta}} & {0 \leq \theta \leq \pi}\end{array} $$

Area of a Region In Exercises \(57-60\) , use the integration capabilities of a graphing utility to approximate, to two decimal places, the area of the region bounded by the graph of the polar equation. $$ r=\frac{9}{4+\cos \theta} $$

Finding the Arc Length of a Polar Curve In Exercises \(57-62,\) use a graphing utility to graph the polar equation over the given interval. Use the integration capabilities of the graphing utility to approximate the length of the curve accurate to two decimal places. $$ \begin{array}{ll}{\text { Polar Equation }} & {\text { Interval }} \\ {r=\sec \theta} & {0 \leq \theta \leq \frac{\pi}{3}} \\\\\end{array} $$

Sketching and Identifying a Conic In Exercises \(13-22\) , find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. $$ r=\frac{-6}{3+7 \sin \theta} $$

Conic What conic section does the polar equation \(r=a \sin \theta+b \cos \theta\) represent?
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