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Precalculus : Express Complex Numbers In Rectangular Form

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Example Questions

Example Question #1 : Express Complex Numbers In Rectangular Form

Convert the following to rectangular form:

$\displaystyle 2(cos(30)+ isin(30))$

Possible Answers:

$2\sqrt{3}+1$ $\displaystyle 2\sqrt{3}+1$

$\sqrt{3}-1$ $\displaystyle \sqrt{3}-1$

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

$\sqrt{3}+i$ $\displaystyle \sqrt{3}+i$

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

Correct answer:

$\sqrt{3}+i$ $\displaystyle \sqrt{3}+i$

Explanation:

Distribute the coefficient 2, and evaluate each term:

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

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Example Question #45 : Polar Coordinates And Complex Numbers

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

Distribute the coefficient and simplify:

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

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Example Question #1 : Polar Form Of Complex Numbers

Represent the polar equation:

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

in rectangular form.

Possible Answers:

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

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

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

$z=i+9\pi$ $\displaystyle z=i+9\pi$

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

Correct answer:

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

Explanation:

Using the general form of a polar equation:

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

we find that the value of $\displaystyle r$ is $\displaystyle 3$ and the value of $\theta$ $\displaystyle \theta$ is $\frac{\pi}{6}$ $\displaystyle \frac{\pi}{6}$.

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

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

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

distributing the 3, we obtain the final answer of:

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

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Example Question #1 : Express Complex Numbers In Rectangular Form

Represent the polar equation:

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

in rectangular form.

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

Using the general form of a polar equation:

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

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

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

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

Distributing the 4, we obtain the final answer of:

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

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Example Question #3 : Express Complex Numbers In Rectangular Form

Represent the polar equation:

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

in rectangular form.

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

Using the general form of a polar equation:

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

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

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

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

distributing the 5, we obtain the final answer of:

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

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Example Question #1 : Express Complex Numbers In Rectangular Form

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

Possible Answers:

$\displaystyle 4.854 - 3.527 i$

$\displaystyle - 3.527 + 4.854 i$

$\displaystyle 4.854 + 3.527 i$

$\displaystyle 3.527 + 4.854i$

Correct answer:

$\displaystyle 4.854 + 3.527 i$

Explanation:

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

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

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Example Question #1 : Express Complex Numbers In Rectangular Form

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

Possible Answers:

$\displaystyle 2.819 - 1.026i$

$\displaystyle -2.819 +1.026i$

$\displaystyle 2.819 + 1.026i$

$\displaystyle -1.026 - 2.819i$

$\displaystyle -2.819 - 1.026i$

Correct answer:

$\displaystyle -2.819 - 1.026i$

Explanation:

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

$\displaystyle 3(-0.9397 -0.3420 ) =-2.819 -1.026i$

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Example Question #1 : Express Complex Numbers In Rectangular Form

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

Possible Answers:

$\displaystyle -1.545 - 4.755 i$

$\displaystyle 1.545 - 4.755 i$

$\displaystyle - 4.755+1.545 i$

$\displaystyle 1.545 + 4.755 i$

$\displaystyle 4.755 - 1.545i$

Correct answer:

$\displaystyle 1.545 - 4.755 i$

Explanation:

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

$\displaystyle 5(0.3090 -0.9511i) =1.545 -4.755i$

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Precalculus : Express Complex Numbers In Rectangular Form

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Express Complex Numbers In Rectangular Form

Example Question #45 : Polar Coordinates And Complex Numbers

Example Question #1 : Polar Form Of Complex Numbers

Example Question #1 : Express Complex Numbers In Rectangular Form

Example Question #3 : 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

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