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Problem 975

# Convert the equation $$\mathrm{r}=\tan \theta+\cot \theta$$ to an equation in cartesian coordinates.

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The short answer is: The given polar equation $$\mathrm{r}=\tan \theta+\cot \theta$$ can be converted to the Cartesian coordinates equation $$(x^2 + y^2)^{3/2} = xy$$.
See the step by step solution

## Step 1: Rewrite the equation in terms of sine and cosine functions

The given equation is: $$\mathrm{r}=\tan \theta+\cot \theta$$ Since we know that: $$\tan \theta = \frac{\sin \theta}{\cos \theta}$$ and $$\cot \theta = \frac{\cos \theta}{\sin \theta}$$, The equation becomes: $$\mathrm{r}=\frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta}$$

## Step 2: Find a common denominator and simplify the equation

To simplify the equation, we will find a common denominator for the fractions, which is $$\cos \theta \sin \theta$$. The equation now becomes: $$\mathrm{r} = \frac{\sin^2 \theta + \cos^2 \theta}{\cos \theta \sin \theta}$$ As we know, $$\sin^2 \theta + \cos^2 \theta = 1$$, so the equation simplifies to: $$\mathrm{r} = \frac{1}{\cos \theta \sin \theta}$$

## Step 3: Substitute the polar coordinates with Cartesian coordinates

Now, we will use the polar to Cartesian coordinate transformations: $$x = r \cos \theta$$ $$y = r \sin \theta$$ We already have the equation: $$\mathrm{r} = \frac{1}{\cos \theta \sin \theta}$$ Substitute $$\cos \theta$$ with $$\frac{x}{r}$$ and $$\sin \theta$$ with $$\frac{y}{r}$$: $$\mathrm{r} = \frac{1}{\frac{x}{r} \cdot \frac{y}{r}}$$

## Step 4: Simplify the equation and solve for $$r$$

Now, simplify the equation: $$\mathrm{r} = \frac{1}{\frac{xy}{r^2}}$$ $$\mathrm{r} = \frac{r^2}{xy}$$ Now, multiply both sides by $$xy$$: $$r^3 = x * y$$

## Step 5: Substitute back Cartesian coordinates for $$r$$

Finally, replace $$r$$ with $$\sqrt{x^2 + y^2}$$ (since $$r^2 = x^2 + y^2$$): $$(\sqrt{x^2 + y^2})^3 = xy$$ This is the Cartesian equation equivalent to the given polar equation: $$x^2 + y^2)^{3/2} = xy$$

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