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Chapter 6: Problem 91

Explain how to convert a point from polar to rectangular coordinates. Provide an example with your explanation.

Short Answer

Expert verified
The steps to convert a point from polar to rectangular coordinates involve understanding and applying the conversion formulas \(x = r\cos\theta\) and \(y = r\sin\theta\). For example, the polar coordinates (3,\(\frac{\pi}{6}\)) convert to rectangular coordinates (\(\frac{3\sqrt{3}}{2}\), \(\frac{3}{2}\)).

Step by step solution

01

Understand the formulas

The formulas for the conversion are derived from the Pythagorean theorem and the concept of the unit circle. \[ x = r\cos\theta \] \[ y = r\sin\theta \] where r is the distance from the origin, \(\theta\) is the angle, x is the coordinate on the horizontal axis and y is the coordinate on the vertical axis.
02

Apply the conversion formulas to an example

Let's take a point in polar coordinates (3,\(\frac{\pi}{6}\)). This means our r = 3 and our \(\theta = \frac{\pi}{6}\). Plug these values into the conversion formulas: \[x = 3\cos{\frac{\pi}{6}}\] \[y = 3\sin{\frac{\pi}{6}}\] Now, calculate x and y using the cosine and sine of \(\frac{\pi}{6}\). This would be respectively, \(\frac{\sqrt{3}}{2}\) and \(\frac{1}{2}\). So, we get: \[x = 3 * \frac{\sqrt{3}}{2}\] \[y = 3 * \frac{1}{2}\] Now simplify x and y. This gives us the rectangular coordinates for the example.
03

Finalize the conversion

After doing the multiplication, we have: \[x = \frac{3\sqrt{3}}{2}\] \[y = \frac{3}{2}\] Thus, the rectangular coordinates equivalent to the polar coordinates (3,\(\frac{\pi}{6}\)) are (\(\frac{3\sqrt{3}}{2}\), \(\frac{3}{2}\)).

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Most popular questions from this chapter

Determine whether v and w are parallel, orthogonal, or neither. $$\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}, \quad \mathbf{w}=6 \mathbf{i}+10 \mathbf{j}$$

Find the angle between \(\mathbf{v}\) and \(\mathbf{w} .\) Round to the nearest tenth of a degree. $$\mathbf{v}=-3 \mathbf{i}+2 \mathbf{j}, \quad \mathbf{w}=4 \mathbf{i}-\mathbf{j}$$

Describe how to find the angle between two vectors.

Find the angle between \(\mathbf{v}\) and \(\mathbf{w} .\) Round to the nearest tenth of a degree. $$\mathbf{v}=-2 \mathbf{i}+5 \mathbf{j}, \quad \mathbf{w}=3 \mathbf{i}+6 \mathbf{j}$$

Find \(\text {pro}_{\mathbf{w}} \mathbf{V}\) Then decompose v into two vectors, \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2},\) where \(\mathbf{v}_{1}\) is parallel to \(\mathbf{w}\) and \(\mathbf{v}_{2}\) is orthogonal to \(\mathbf{w}.\) $$\mathbf{v}=\mathbf{i}+3 \mathbf{j}, \quad \mathbf{w}=-2 \mathbf{i}+5 \mathbf{j}$$
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