# Harrison's question via Facebook about polar functions

• MHB
• Prove It
In summary, the two given curves have two points, \displaystyle \begin{align*} \left( \rho , \alpha \right) \end{align*} and \displaystyle \begin{align*} \left( \rho , \beta \right) \end{align*}, which are the same distance from the origin and satisfy the conditions \displaystyle \begin{align*} 0 \leq \theta \leq \pi \end{align*} and \displaystyle \begin{align*} \sqrt{3}\,\pi \end{align*} between them. It is shown that \displaystyle \begin{align*} \alpha \end{align Prove It Gold Member MHB The point\displaystyle \begin{align*} \left( \rho , \alpha \right) \end{align*}$lies on the curve$\displaystyle \begin{align*} r = \frac{3\,\theta}{2} \end{align*}$and the point$\displaystyle \begin{align*} \left( \rho , \beta \right) \end{align*}$lies on the curve$\displaystyle \begin{align*} r = \theta + \pi \end{align*}$, such that the points are the same distance from the origin,$\displaystyle \begin{align*} 0\leq \theta \leq \pi \end{align*}$and the distance between them is$\displaystyle \begin{align*} \sqrt{3}\,\pi \end{align*}$. Show that$\displaystyle \begin{align*} \alpha \end{align*}$satisfies$\displaystyle \begin{align*} \frac{2\,\pi^2}{3} = \alpha ^2 \,\left[ 1 + \cos{ \left( \frac{\alpha}{2} \right) } \right] \end{align*}$Since the distances from the origin$\displaystyle \begin{align*} \rho \end{align*}$are the same, we can say$\displaystyle \begin{align*} \rho = \frac{3\,\alpha}{2} \end{align*}$and$\displaystyle \begin{align*} \rho = \beta + \pi \end{align*}$, giving$\displaystyle \begin{align*} \frac{3\,\alpha}{2} &= \beta + \pi \\ \beta &= \frac{3\,\alpha}{2} - \pi \end{align*}$The distance between two points in polar form$\displaystyle \begin{align*} \left( r_1 , \theta_1 \right) \end{align*}$and$\displaystyle \begin{align*} \left( r_2, \theta_2 \right) \end{align*}$is given by$\displaystyle \begin{align*} d = \sqrt{r_1^2 + r_2^2 - 2\,r_1\,r_2\,\cos{ \left( \theta_1 - \theta_2 \right) }} \end{align*}$, so in this case$\displaystyle \begin{align*} \sqrt{3}\,\pi &= \sqrt{ \rho^2 + \rho^2 - 2\,\rho^2 \,\cos{ \left( \alpha - \beta \right) } } \\ \sqrt{3}\,\pi &= \sqrt{ 2\,\rho^2 \,\left[ 1 - \cos{ \left( \alpha - \beta \right) } \right] } \\ 3\,\pi^2 &= 2\,\rho ^2 \,\left[ 1 - \cos{ \left( \alpha - \beta \right) } \right] \\ 3\,\pi^2 &= 2\,\left( \frac{3\,\alpha}{2} \right) ^2 \,\left\{ 1 - \cos{ \left[ \alpha - \left( \frac{3\,\alpha}{2} - \pi \right) \right] } \right\} \\ 3\,\pi^2 &= \frac{9\,\alpha^2}{2} \,\left[ 1 - \cos{\left( \pi - \frac{\alpha}{2} \right) } \right] \\ \frac{2\,\pi^2}{3} &= \alpha^2 \,\left[ 1 - \cos{\left( \pi - \frac{\alpha}{2} \right) } \right] \\ \frac{2\,\pi^2}{3} &= \alpha^2\,\left[ 1 + \cos{ \left( \frac{\alpha}{2} \right) } \right] \end{align*}\$

benorin
QED

benorin

## 1. What are polar functions?

Polar functions are mathematical functions that are expressed in terms of polar coordinates, which use a distance from a fixed point and an angle from a fixed direction to specify a point in a plane.

## 2. How are polar functions different from Cartesian functions?

Polar functions use polar coordinates, while Cartesian functions use rectangular coordinates. This means that polar functions describe points in terms of distance and angle, while Cartesian functions describe points in terms of x and y coordinates.

## 3. What are the advantages of using polar functions?

Polar functions can be useful in situations where the distance and angle from a fixed point are more relevant than the x and y coordinates. They can also simplify certain mathematical equations and make them easier to visualize.

## 4. How do you graph a polar function?

To graph a polar function, you first need to convert the polar coordinates into Cartesian coordinates. Then, plot the points on a graph and connect them to create the shape of the function. You can also use a graphing calculator or online graphing tool to help with this process.

## 5. What are some common examples of polar functions?

Some common examples of polar functions include circles, cardioids, and limaçons. They are also commonly used in physics and engineering to describe the motion of objects in circular or rotational patterns.

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