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Precalculus : Polar Form of Complex Numbers

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Example Question #1 :Express Complex Numbers In Rectangular Form

Convert the following to rectangular form:

2(cos(30)+ isin(30))

Possible Answers:

$2\sqrt{3}+1$

$\sqrt{3}+2i$

$2\sqrt{3}+2i$

$\sqrt{3}-1$

$\sqrt{3}+i$

Correct answer:

$\sqrt{3}+i$

Explanation:

Distribute the coefficient 2, and evaluate each term:

$\\2(cos(30)+ isin(30))\\ \\= 2cos(30)+i(2sin(30))\\ \\=\sqrt{3}+i$

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Example Question #1 :Express Complex Numbers In Rectangular Form

Convert the following to rectangular form: 3(cos(60)-isin(30))

Possible Answers:

$\frac{3}{2}-\frac{i}{2}$

$\frac{\sqrt3}{2}-\frac{i}{2}$

$\frac{3\sqrt3}{2}-\frac{i}{2}$

$\frac{3}{2}-\frac{3}{2}i$

$\frac{3}{2}-\frac{\sqrt3 i}{2}$

Correct answer:

$\frac{3}{2}-\frac{3}{2}i$

Explanation:

Distribute the coefficient and simplify:

$\\ 3(cos(60)-isin(30))\\ \\=3sin(60)-i(3sin(30)))\\ \\=\frac{3}{2}-\frac{3}{2}i$

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Example Question #3 :Express Complex Numbers In Rectangular Form

Represent the polar equation:

$z = 3\left(cos(\frac{\pi}{6} )+i\cdot sin\left(\frac{\pi}{6}\right)\right)$

in rectangular form.

Possible Answers:

$z=i+9\pi$

$z = \frac{3\sqrt{3}}{2} + \frac{3}{2}i$

$z = \frac{2\sqrt{2}}{3} + \frac{2}{3}i$

$z = cos\left(\frac{\pi}{6}\right) + sin\left(\frac{\pi}{6}\right)$

$z = 3cos\left(\frac{\pi}{6}\right) + 3sin\left(\frac{\pi}{6}\right)$

Correct answer:

$z = \frac{3\sqrt{3}}{2} + \frac{3}{2}i$

Explanation:

Using the general form of a polar equation:

$z = r(cos(\theta) + i\cdot sin(\theta))$

we find that the value ofisand the value of $\theta$ is $\frac{\pi}{6}$ .

The rectangular form of the equation appears as a + bi , and can be found by finding the trigonometric values of the cosine and sine equations.

$cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$

$sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$

distributing the 3, we obtain the final answer of:

$\frac{3\sqrt{3}}{2} + \frac{3}{2}i$

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Example Question #4 :Express Complex Numbers In Rectangular Form

Represent the polar equation:

$z = 4\left(cos\left(\frac{\pi}{4}\right) + i\cdot sin\left(\frac{\pi}{4}\right)\right)$

in rectangular form.

Possible Answers:

$\sqrt{2} + 2i$

$2 + \sqrt{2}i$

$2\sqrt{2} + 2\sqrt{2}i$

$4cos\left(\frac{\pi}{4}\right) + 4sin\left(\frac{\pi}{4}\right)$

$cos\left(\frac{\pi}{4}\right) + sin\left(\frac{\pi}{4}\right)i$

Correct answer:

$2\sqrt{2} + 2\sqrt{2}i$

Explanation:

Using the general form of a polar equation:

$z = r(cos(\theta) + i\cdot sin(\theta))$

we find that the value of r=4 and the value of $\theta =\frac{\pi}{4}$ . The rectangular form of the equation appears as a + bi , and can be found by finding the trigonometric values of the cosine and sine equations.

$cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}$

$sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt2}$

Distributing the 4, we obtain the final answer of:

$z = 2\sqrt{2} + 2\sqrt{2}i$

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Example Question #1 :Express Complex Numbers In Rectangular Form

Represent the polar equation:

$z = 5\left(cos\left(\frac{3\pi}{4}\right)+i \cdot sin\left(\frac{3\pi}{4}\right)\right)$

in rectangular form.

Possible Answers:

$z = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i$

$z = \frac{-5\sqrt{2}}{2} + \frac{5\sqrt{2}}{2}i$

$z = \frac{-5}{2} + \frac{5}{2}i$

$z = \frac{5\sqrt{2}}{2} + \frac{-5\sqrt{2}}{2}i$

$z = cos(\frac{3\pi}{4}) + sin(\frac{3\pi}{4})i$

Correct answer:

$z = \frac{-5\sqrt{2}}{2} + \frac{5\sqrt{2}}{2}i$

Explanation:

Using the general form of a polar equation:

$z = r(cos(\theta) + i*sin(\theta))$

we find that the value of r=5 and the value of $\theta =\frac{3\pi}{4}$ . The rectangular form of the equation appears as a + bi , and can be found by finding the trigonometric values of the cosine and sine equations.

$cos\left(\frac{3\pi}{4}\right) = \frac{-1}{\sqrt{2}}$

$sin\left(\frac{3\pi}{4}\right) = \frac{1}{\sqrt{2}}$

distributing the 5, we obtain the final answer of:

$\frac{-5\sqrt{2}}{2} + \frac{5\sqrt{2}}{2}i$

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Example Question #1 :Express Complex Numbers In Rectangular Form

Convert $6(\cos \frac{\pi}{5} + i \sin \frac{ \pi }{5} )$ in rectangular form

Possible Answers:

4.854 + 3.527 i

3.527 + 4.854i

- 3.527 + 4.854 i

4.854 - 3.527 i

Correct answer:

4.854 + 3.527 i

Explanation:

To convert, just evaluate the trig ratios and then distribute the radius.

6(0.8090 + i 0. 5878) = 4.854+ 3.527i

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Example Question #1 :Express Complex Numbers In Rectangular Form

Convert $3(\cos 200^o + i \sin 200^o )$ to rectangular form

Possible Answers:

2.819 - 1.026i

-2.819 - 1.026i

-1.026 - 2.819i

2.819 + 1.026i

-2.819 +1.026i

Correct answer:

-2.819 - 1.026i

Explanation:

To convert to rectangular form, just evaluate the trig functions and then distribute the radius:

3(-0.9397 -0.3420 ) =-2.819 -1.026i

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Example Question #1 :Express Complex Numbers In Rectangular Form

Convert $5(\cos \frac{ 8 \pi }{5} + i \sin \frac{ 8 \pi }{5} )$ to rectangular form

Possible Answers:

1.545 + 4.755 i

1.545 - 4.755 i

- 4.755+1.545 i

4.755 - 1.545i

-1.545 - 4.755 i

Correct answer:

1.545 - 4.755 i

Explanation:

To convert, evaluate the trig ratios and then distribute the radius:

5(0.3090 -0.9511i) =1.545 -4.755i

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Example Question #1 :Express Complex Numbers In Polar Form

The following equation has complex roots:

x^2 - 3x + 9 = 0

Express these roots in polar form.

Possible Answers:

$\sqrt{3}(\cos 30 \pm i \sin 30)$

$\sqrt{3}(\cos 60 \pm i\sin 60)$

$3(\cos 30 \pm i\sin(30))$

$3(\cos 60 \pm i\sin 60)$

Correct answer:

$3(\cos 60 \pm i\sin 60)$

Explanation:

Every complex number can be written in the forma + bi

The polar form of a complex number takes the form r(cos $\Theta$ + isin $\Theta$ )

Now r can be found by applying the Pythagorean Theorem on a and b, or:

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

$\Theta$ can be found using the formula:

$\Theta$ = $tan^{-1}(b/a)$

So for this particular problem, the two roots of the quadratic equation

x^2 - 3x + 9 = 0

are:

$x = \frac{3\pm 3\sqrt{3}i}{2}$

Hence, a = 3/2 and b = 3√3 / 2

Therefore r = $\sqrt{a^{2} + b^{2}}$ =3

and $\Theta$ = tan^-1 (√3) = 60

And therefore x = r(cos $\Theta$ + isin $\Theta$ ) = 3 (cos 60 + isin 60)

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Example Question #2 :Express Complex Numbers In Polar Form

Express the roots of the following equation in polar form.

x^2+3x+3

Possible Answers:

$\sqrt{3}(\cos(30)+i\sin(30))$

$(\cos(150)+i\sin(150))$

$\sqrt{3}(\cos(150)+i\sin(150))$

$\sqrt{3}(\cos(-30)+i\sin(-30))$

$3(\cos(150)+i\sin(150))$

Correct answer:

$\sqrt{3}(\cos(150)+i\sin(150))$

Explanation:

First, we must use the quadratic formula to calculate the roots in rectangular form.

$\frac{-3\pm \sqrt{9-36}}{2}=\frac{-3\pm i\sqrt{3}}{2}$

Remembering that the complex roots of the equation take on the form a+bi,

we can extract the a and b values.

$a=-\frac{3}{2}$

$b=\frac{\sqrt{3}}{2}$

We can now calculate r and theta.

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

$\theta=\tan^{-1}\frac{b}{a}$

Using these two relations, we get

$r=\sqrt{3}$

$\theta=-30$ . 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.

The angle theta now becomes 150.

$\theta=150$ .

You can now plug in r and theta into the standard polar form for a number:

$r(\cos(\theta) +i\sin(\theta ))$

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Precalculus : Polar Form of Complex Numbers

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 :Express Complex Numbers In Rectangular Form

Example Question #1 :Express Complex Numbers In Rectangular Form

Example Question #3 :Express Complex Numbers In Rectangular Form

Example Question #4 :Express Complex Numbers In Rectangular Form

Example Question #1 :Express Complex Numbers In Rectangular Form

Example Question #1 :Express Complex Numbers In Rectangular Form

Example Question #1 :Express Complex Numbers In Rectangular Form

Example Question #1 :Express Complex Numbers In Rectangular Form

Example Question #1 :Express Complex Numbers In Polar Form

Example Question #2 :Express Complex Numbers In Polar Form

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