# How to express complex numbers in polar form

Polar Form of a Complex Number

The following equation has complex roots: Express these roots in polar form. Possible Answers: Correct answer: Explanation: Every complex number can be written in the form a + bi. The polar form of a complex number takes the form r (cos + isin) Now r can be . Writing Complex Numbers in Polar Form The polar form of a complex number expresses a number in terms of an angle ? and its distance from the origin r. Given a complex number in rectangular form expressed as z = x + yi, we use the same conversion formulas as we do to write the number in trigonometric form: x = rcos? y = rsin? r = vx2 + y2.

Converting Complex Numbers to Polar Form :. Here we are going to see some example problems based on converting complex numbers to polar form.

The polar form or trigonometric form of a complex number P is. The value "r" represents the absolute value or modulus of the complex number z. To find the principal argument of a complex number, we may use the following methods. The capital A is important here to distinguish the principal value from the general value. Question 1 :. Solution :. After having gone through the stuff given above, we hope that the students would have understood, " Converting Complex Numbers to Polar Form". Apart from the stuff given in this section " Converting Complex Numbers to Polar Form"if you need any other stuff in math, please use our google custom search here.

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Output Type

Polar form of a complex number shown on a complex plane. Figure shows how the rectangular and polar forms are related. There we have plotted the complex number a + bi. If we now connect that point to the origin by a line segment of length r, it makes an angle ? with the horizontal axis. Answer. A complex number ?? is said to be in polar form if ?? = ?? (?? + ?? ??), c o s s i n where ?? is the modulus of ?? and ?? is the argument. We will consider each option in turn and see whether it is correctly expressed in this form. Initially, this number looks like it is in polar form. Convert Complex Numbers to Polar Form. Added May 14, by mrbartonmaths in Mathematics. convert complex numbers to polar co-ordinates.

However, we need to adjust this theta to reflect the real location of the vector, which is in the 2nd quadrant a is negative, b is positive ; a represents the x-axis in the real-imaginary plane, b represents the y-axis. Express the complex number in polar form. The figure below shows a complex number plotted on the complex plane. The horizontal axis is the real axis and the vertical axis is the imaginary axis. The polar form of a complex number is. We want to find the real and complex components in terms of and where is the length of the vector and is the angle made with the real axis.

We use the Pythagorean Theorem to find :. We find by solving the trigonometric ratio. Using ,. Then we plug and into our polar equation to obtain. What is the polar form of the complex number? The polar form of a complex number is where is the modulus of the complex number and is the angle in radians between the real axis and the line that passes through and. We can solve for and easily for the complex number :. Remember that the standard form of a complex number is: , which can be rewritten in polar form as:.

To find r, we must find the length of the line by using the Pythagorean theorem:. To find , we can use the equation. Given these identities, first solve for and. The polar form of a complex number is:. Convert to polar form:. First, find the radius :. Then find the angle, thinking of the imaginary part as the height and the radius as the hypotenuse of a right triangle:.

We can get the positive coterminal angle by adding :. First find the radius, :. Now find the angle, thinking of the imaginary part as the height and the radius as the hypotenuse of a right triangle:. The complex number in polar form is. Convert the complex number to polar form. First find :.

Now find the angle. Consider the imaginary part to be the height of a right triangle with hypotenuse. What the calculator does not know is that this angle is actually located in quadrant II, since the real part is negative and the imaginary part is positive.

To find the angle in quadrant II whose sine is also , subtract from :. If you've found an issue with this question, please let us know. With the help of the community we can continue to improve our educational resources. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

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Correct answer:. Report an Error. Express the roots of the following equation in polar form. Explanation : First, we must use the quadratic formula to calculate the roots in rectangular form. We can now calculate r and theta. Using these two relations, we get. The angle theta now becomes You can now plug in r and theta into the standard polar form for a number:. Explanation : The figure below shows a complex number plotted on the complex plane.

We use the Pythagorean Theorem to find : We find by solving the trigonometric ratio Using , Then we plug and into our polar equation to obtain. Explanation : The correct answer is The polar form of a complex number is where is the modulus of the complex number and is the angle in radians between the real axis and the line that passes through and.

We can solve for and easily for the complex number : which gives us. Express the complex number in polar form:. Explanation : Remember that the standard form of a complex number is: , which can be rewritten in polar form as:.

To find r, we must find the length of the line by using the Pythagorean theorem: To find , we can use the equation Note that this value is in radians, NOT degrees. Thus, the polar form of this equation can be written as. Express this complex number in polar form. Explanation : Given these identities, first solve for and. The polar form of a complex number is: at because the original point, 1,1 is in Quadrant 1 Therefore Explanation : First, find the radius : Then find the angle, thinking of the imaginary part as the height and the radius as the hypotenuse of a right triangle: according to the calculator.

We can get the positive coterminal angle by adding : The polar form is. Explanation : First find the radius, : Now find the angle, thinking of the imaginary part as the height and the radius as the hypotenuse of a right triangle: according to the calculator.

This is an appropriate angle to stay with since this number should be in quadrant I. Explanation : First find : Now find the angle. To find the angle in quadrant II whose sine is also , subtract from : The complex number in polar form is.

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