Express Complex Numbers In Rectangular Form - Pre-Calculus

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Question

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

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Answer

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

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Question

Convert the following to rectangular form:

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Answer

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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Question

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

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Answer

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

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Question

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

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Answer

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

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Question

Represent the polar equation:

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

in rectangular form.

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Answer

Using the general form of a polar equation:

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

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

The rectangular form of the equation appears as , 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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Question

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.

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Answer

Using the general form of a polar equation:

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

we find that the value of and the value of $\theta =\frac{\pi}{4}$$ . The rectangular form of the equation appears as , 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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Question

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.

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Answer

Using the general form of a polar equation:

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

we find that the value of and the value of $\theta =\frac{3\pi}{4}$$ . The rectangular form of the equation appears as , 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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Question

Convert the following to rectangular form:

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Answer

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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Question

Represent the polar equation:

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

in rectangular form.

Tap to reveal answer

Answer

Using the general form of a polar equation:

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

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

The rectangular form of the equation appears as , 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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Question

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.

Tap to reveal answer

Answer

Using the general form of a polar equation:

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

we find that the value of and the value of $\theta =\frac{\pi}{4}$$ . The rectangular form of the equation appears as , 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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Question

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.

Tap to reveal answer

Answer

Using the general form of a polar equation:

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

we find that the value of and the value of $\theta =\frac{3\pi}{4}$$ . The rectangular form of the equation appears as , 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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Question

Convert the following to rectangular form:

Tap to reveal answer

Answer

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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Question

Convert the following to rectangular form:

Tap to reveal answer

Answer

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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Question

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

Tap to reveal answer

Answer

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

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Question

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

Tap to reveal answer

Answer

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

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Question

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

Tap to reveal answer

Answer

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

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Question

Represent the polar equation:

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

in rectangular form.

Tap to reveal answer

Answer

Using the general form of a polar equation:

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

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

The rectangular form of the equation appears as , 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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Question

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.

Tap to reveal answer

Answer

Using the general form of a polar equation:

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

we find that the value of and the value of $\theta =\frac{\pi}{4}$$ . The rectangular form of the equation appears as , 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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Question

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.

Tap to reveal answer

Answer

Using the general form of a polar equation:

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

we find that the value of and the value of $\theta =\frac{3\pi}{4}$$ . The rectangular form of the equation appears as , 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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Question

Convert the following to rectangular form:

Tap to reveal answer

Answer

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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Knew It

Express Complex Numbers In Rectangular Form - Pre-Calculus

Convert to rectangular form

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

Convert the following to rectangular form:

Distribute the coefficient and simplify:

Convert in rectangular form

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

Convert to rectangular form

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

Represent the polar equation: in rectangular form.

Using the general form of a polar equation: we find that the value of is and the value of is . The rectangular form of the equation appears as , and can be found by finding the trigonometric values of the cosine and sine equations. distributing the 3, we obtain the final answer of:

Represent the polar equation: in rectangular form.

Using the general form of a polar equation: we find that the value of and the value of . The rectangular form of the equation appears as , and can be found by finding the trigonometric values of the cosine and sine equations. Distributing the 4, we obtain the final answer of:

Represent the polar equation: in rectangular form.

Using the general form of a polar equation: we find that the value of and the value of . The rectangular form of the equation appears as , and can be found by finding the trigonometric values of the cosine and sine equations. distributing the 5, we obtain the final answer of:

Convert the following to rectangular form:

Distribute the coefficient 2, and evaluate each term:

Represent the polar equation: in rectangular form.

Using the general form of a polar equation: we find that the value of is and the value of is . The rectangular form of the equation appears as , and can be found by finding the trigonometric values of the cosine and sine equations. distributing the 3, we obtain the final answer of:

Represent the polar equation: in rectangular form.

Using the general form of a polar equation: we find that the value of and the value of . The rectangular form of the equation appears as , and can be found by finding the trigonometric values of the cosine and sine equations. Distributing the 4, we obtain the final answer of:

Represent the polar equation: in rectangular form.

Using the general form of a polar equation: we find that the value of and the value of . The rectangular form of the equation appears as , and can be found by finding the trigonometric values of the cosine and sine equations. distributing the 5, we obtain the final answer of:

Convert the following to rectangular form:

Distribute the coefficient 2, and evaluate each term:

Convert the following to rectangular form:

Distribute the coefficient and simplify:

Convert in rectangular form

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

Convert to rectangular form

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

Convert to rectangular form

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

Represent the polar equation: in rectangular form.

Using the general form of a polar equation: we find that the value of is and the value of is . The rectangular form of the equation appears as , and can be found by finding the trigonometric values of the cosine and sine equations. distributing the 3, we obtain the final answer of:

Represent the polar equation: in rectangular form.

Using the general form of a polar equation: we find that the value of and the value of . The rectangular form of the equation appears as , and can be found by finding the trigonometric values of the cosine and sine equations. Distributing the 4, we obtain the final answer of:

Represent the polar equation: in rectangular form.

Using the general form of a polar equation: we find that the value of and the value of . The rectangular form of the equation appears as , and can be found by finding the trigonometric values of the cosine and sine equations. distributing the 5, we obtain the final answer of:

Convert the following to rectangular form:

Distribute the coefficient 2, and evaluate each term:

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

Distribute the coefficient and simplify:

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

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

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

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.

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.

Distribute the coefficient 2, and evaluate each term:

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

Represent the polar equation:

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

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.

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.

Distribute the coefficient 2, and evaluate each term:

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

Distribute the coefficient and simplify:

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

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

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

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

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.

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.

Distribute the coefficient 2, and evaluate each term:

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