Rewriting 12 in Polar Form

Oct 13, 2025 by magicalceylontours.com 117 views

Rewrite in Polar Form: Understanding the Basics

Polar form is a way of expressing complex numbers in terms of their distance from the origin and the angle they make with the positive x-axis. In this article, we will explore how to rewrite a given number in polar form, specifically focusing on the number 12.

What is Polar Form?

Polar form is a method of representing complex numbers in the form $r(\cos \theta + i\sin \theta)$ , where $r$ is the distance from the origin to the point, and $\theta$ is the angle between the positive x-axis and the line segment connecting the origin to the point. This form is particularly useful for simplifying complex number operations, such as multiplication and division.

Rewriting 12 in Polar Form

To rewrite 12 in polar form, we need to find the distance from the origin to the point (12, 0) and the angle it makes with the positive x-axis. Since 12 is a real number, it lies on the x-axis, and the angle it makes with the positive x-axis is 0.

Option A: $r=12 \cos \theta$

Using the formula for polar form, we can rewrite 12 as $r=12 \cos \theta$ . However, this is not the correct polar form of 12, as the cosine function is not defined for $\theta = 0$ . In polar form, the cosine function is typically used to represent the x-coordinate of a point, which is not the case here.

Option B: $r=12 \sin \theta$

Similarly, we can rewrite 12 as $r=12 \sin \theta$ . However, this is also not the correct polar form of 12, as the sine function is not defined for $\theta = 0$ . In polar form, the sine function is typically used to represent the y-coordinate of a point, which is not the case here.

Option C: $r=12 \sec \theta$

Using the formula for polar form, we can rewrite 12 as $r=12 \sec \theta$ . However, this is not the correct polar form of 12, as the secant function is not defined for $\theta = 0$ . In polar form, the secant function is typically used to represent the reciprocal of the cosine function, which is not the case here.

Option D: $r=12 \csc \theta$

Similarly, we can rewrite 12 as $r=12 \csc \theta$ . However, this is also not the correct polar form of 12, as the cosecant function is not defined for $\theta = 0$ . In polar form, the cosecant function is typically used to represent the reciprocal of the sine function, which is not the case here.

The Correct Polar Form of 12

The correct polar form of 12 is simply $r=12$ . This is because 12 is a real number, and it lies on the x-axis, making an angle of 0 with the positive x-axis. In polar form, the distance from the origin to the point is represented by the variable $r$ , and the angle it makes with the positive x-axis is represented by the variable $\theta$ . Since 12 lies on the x-axis, the angle $\theta$ is 0, and the distance $r$ is simply 12.

Conclusion

In conclusion, rewriting 12 in polar form is a straightforward process that involves finding the distance from the origin to the point and the angle it makes with the positive x-axis. The correct polar form of 12 is simply $r=12$ , as it lies on the x-axis, making an angle of 0 with the positive x-axis.
Polar Form Q&A: Frequently Asked Questions

In our previous article, we explored the basics of polar form and how to rewrite a given number in polar form. In this article, we will answer some frequently asked questions about polar form to help you better understand this concept.

Q: What is the difference between polar form and rectangular form?

A: Rectangular form is a way of representing complex numbers in the form $a+bi$ , where $a$ and $b$ are real numbers. Polar form, on the other hand, is a way of representing complex numbers in the form $r(\cos \theta + i\sin \theta)$ , where $r$ is the distance from the origin to the point and $\theta$ is the angle it makes with the positive x-axis.

Q: How do I convert a complex number from rectangular form to polar form?

A: To convert a complex number from rectangular form to polar form, you need to find the distance from the origin to the point and the angle it makes with the positive x-axis. You can use the following formulas:

$r = \sqrt{a^2 + b^2}$ $\theta = \tan^{-1}\left(\frac{b}{a}\right)$

Q: How do I convert a complex number from polar form to rectangular form?

A: To convert a complex number from polar form to rectangular form, you need to use the following formulas:

$a = r\cos \theta$ $b = r\sin \theta$

Q: What is the advantage of using polar form?

A: Polar form has several advantages over rectangular form. It is particularly useful for simplifying complex number operations, such as multiplication and division. Additionally, polar form is often used in engineering and physics applications, where it is easier to work with distances and angles rather than rectangular coordinates.

Q: Can I use polar form with negative numbers?

A: Yes, you can use polar form with negative numbers. However, you need to be careful when dealing with negative numbers in polar form, as the distance from the origin to the point may be negative. In this case, you need to use the following formula:

$r = -\sqrt{a^2 + b^2}$

Q: Can I use polar form with imaginary numbers?

A: Yes, you can use polar form with imaginary numbers. However, you need to be careful when dealing with imaginary numbers in polar form, as the angle it makes with the positive x-axis may be different from the angle it makes with the positive y-axis. In this case, you need to use the following formula:

$\theta = \tan^{-1}\left(\frac{b}{a}\right) + \pi$

Q: How do I graph a complex number in polar form?

A: To graph a complex number in polar form, you need to plot the point on the complex plane using the distance from the origin to the point and the angle it makes with the positive x-axis. You can use the following formulas to find the x and y coordinates of the point:

$x = r\cos \theta$ $y = r\sin \theta$

Conclusion

In conclusion, polar form is a powerful tool for working with complex numbers. By understanding the basics of polar form and how to convert between rectangular and polar form, you can simplify complex number operations and make it easier to work with distances and angles. We hope this Q&A article has helped you better understand polar form and how to use it in your math and science applications.