Tagged Questions

Questions on polar coordinates, a coordinate system where points are represented by their distance from the origin ($r$) and the angle the line joining the point and the origin makes with the positive horizontal axis ($\theta$).

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Polar Equation to Rectangular

$$r=\frac{9}{4 \cos θ − 3 \sin θ}$$ How do I do this? (Equation is in polar form.) I have already tried to do this, but I don't know how to finish it.
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Help needed on evaluating the following double integral

Use polar coordinates to evaluate $\int\int_{D}\ x\ dA$ where $D$ is the region inside the circle $x^2 + (y-1)^2 = 1$ but outside the circle $x^2 + y^2 =1$. (It's like a crescent moon facing the ...
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how to write floor function vectors in polar coordinates

let $$\lfloor{x}\rfloor=y$$ And $$z=x-\lfloor{x}\rfloor$$ Plot the following vector in polar coordinates: $$x\hat{\imath}+(y/z)\hat{\jmath}$$ I know that while transforming from cartesian to polar we ...
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Find the center of mass of homogeneous object

I am asked to find the center of mass of this homogeneous object: Let's say that it's density is $k$ so the mass is $$m = \int_{0}^{\pi} \int_{a}^{2a} k r drd\theta = \frac{3\pi a^2 k}{2}$$ So ...
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How to convert $\theta = \pi/3$ into cartesian form?

How can I convert $$\theta = \frac{\pi}{3}$$ into cartesian form? What I get is $$\theta = \frac{\pi}{3}\\ cos(\theta) = \frac{x}{r} = \frac{1}{2}\\ x = \frac{r}{2}$$ and I'm not sure what the ...
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Find a Cartesian equation for the curve and identify it. $r^2 \cos 2\theta = 1$

Find a Cartesian equation for the curve and identify it. $$r^2 \cos 2\theta = 1$$ I'm confused by the $2\theta.$ I isolated $r^2$ to get $r^2 = \frac{1}{\cos2\theta}$ Now, normally if it was just ...
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Rotating point by angle

Let $X = (c, 0)$. If I will rotate $X$ by, say, angle $\alpha = \frac{\pi}{4}$, how can I determine position of new angle? Will it just be $X' = (c + \cos\alpha, \sin\alpha)$?
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Convert polar velocity components to Cartesian

I haven't been able to find an answer to velocity component transformation from polar to Cartesian on here, so I'm hoping that someone might be able to answer this question for me. I am given a ...
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Proof of complex numbers

Let $w \in \mathbb C$ where $|w|=1$. I am trying to prove that there exists $\theta \in \mathbb R$ such that $- \frac{i}{2}(w^n-w^{-n})=\sin(n\theta)$ for all $n\in \mathbb N$ To begin, I thought ...
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polar coordinates vector equation of a rectangle

We can write the equation of the circle in vector form in polar coordinates as: $$\vec{r}=R\hat{r}$$ ; where 'R' is the radius of the circle. Similarly, can we write the vector equation for a ...
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How to define the domain of functions in polar coordiante?

I am rather confused by how we should assign the domain (the interval of values of $\theta$) in functions with polar coordinates. To be more specific, in the following image, we should find the ...
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Find the Volume lying inside both the sphere $x^2+y^2+z^2=a^2$ and the cylinder $x^2+y^2=ax$

Taking the equation for the cylinder I completed the square to find $(x-\frac{a}{2})^2+y^2=\frac{a^2}{4}$ and the sphere clearly has radius $a$ and is centered at the origin. Now to solve this ...