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Example Questions
Example Question #3 :Express Complex Numbers In Polar Form
Express the complex numberin 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 ofandwhereis the length of the vector andis the angle made with the real axis.
We use the Pythagorean Theorem to find:
We findby solving the trigonometric ratio
Using,
Then we plugandinto our polar equation to obtain
Example Question #4 :Express Complex Numbers In Polar Form
What is the polar form of the complex number?
The correct answer is
The polar form of a complex numberiswhereis the modulus of the complex number andis the angle in radians between the real axis and the line that passes through(and). We can solve forandeasily for the complex number:
which gives us
Example Question #5 :Express Complex Numbers In Polar Form
Express the complex number in polar form:
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 lineby 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
Example Question #6 :Express Complex Numbers In Polar Form
Express this complex number in polar form.
None of these answers are correct.
Given these identities, first solve forand. The polar form of a complex number is:
at(because the original point, (1,1) is in Quadrant 1)
Therefore...
Example Question #7 :Express Complex Numbers In Polar Form
有限公司nvert 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:
according to the calculator.
We can get the positive coterminal angle by adding:
The polar form is
Example Question #8 :Express Complex Numbers In Polar Form
有限公司nvert to polar form:
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.
The complex number in polar form is
Example Question #9 :Express Complex Numbers In Polar Form
有限公司nvert the complex numberto polar form
First find:
Now find the angle. Consider the imaginary part to be the height of a right triangle with hypotenuse.
according to the calculator.
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:
The complex number in polar form is
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