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Chapter 6: Problem 94

In converting \(r=5\) from a polar equation to a rectangular equation, describe what should be done to both sides of the equation and why this should be done.

Short Answer

Expert verified
The rectangular equation equivalent to the given polar equation \(r=5\) is \(x^2 + y^2 = 25\). This is achieved by using the Pythagorean theorem to create the relationship between polar and rectangular coordinates, and then substituting the value of r.

Step by step solution

01

Start with the given polar equation

The given polar equation is \(r=5\). It represents a circle with a radius of 5 units centered at the origin in polar coordinates.
02

Apply the conversion formula for polar to rectangular coordinates

According to the Pythagorean theorem, the relationship between the polar coordinates r, 胃 and the rectangular coordinates x and y is \(x^2 = r^2 \cdot cos^2(胃)\) and \(y^2 = r^2 \cdot sin^2(胃)\). Also, we know that \(sin^2(胃) + cos^2(胃) = 1\).
03

Combine the equations

Now, combine the previous equations to create one equation only involving x and y. This gives us: \(x^2 + y^2 = r^2\).
04

Substituting the value of r

The value of r for our circle is 5. So after substituting the value of r, we have \(x^2 + y^2 = 25\) which is the equivalent rectangular equation for the given polar equation.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'm graphing a polar equation in which for every value of \(\theta\) there is exactly one corresponding value of \(r,\) yet my polar coordinate graph fails the vertical line for functions.

Explain how to find the dot product of two vectors.

Use the dot product to determine whether v and w are orthogonal. $$\mathbf{v}=\mathbf{i}+\mathbf{j}, \quad \mathbf{w}=-\mathbf{i}+\mathbf{j}$$

Find two vectors \(\mathbf{v}\) and \(\mathbf{w}\) such that the projection of \(\mathbf{v}\) onto \(\mathbf{w}\) is \(\mathbf{v}\).

Let $$\mathbf{u}=-\mathbf{i}+\mathbf{j}, \quad \mathbf{v}=3 \mathbf{i}-2 \mathbf{j}, \quad \text { and } \quad \mathbf{w}=-5 \mathbf{j}$$ Find each specified scalar or vector. $$4 \mathbf{u} \cdot(5 \mathbf{v}-3 \mathbf{w})$$
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