Express Complex Numbers In Rectangular Form

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Pre-Calculus › Express Complex Numbers In Rectangular Form

Questions 1 - 8

1

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

Explanation

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

2

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.

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

$\sqrt{2} + 2i$

$2 + \sqrt{2}i$

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

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

Explanation

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:

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

3

Represent the polar equation:

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

in rectangular form.

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

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

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

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

$z=i+9\pi$

Explanation

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:

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

4

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

Explanation

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

5

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.

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

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

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

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

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

Explanation

Using the general form of a polar equation:

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

we find that the value of 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:

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

6

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

Explanation

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

7

Convert the following to rectangular form:

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

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

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

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

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

Explanation

Distribute the coefficient and simplify:

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

8

Convert the following to rectangular form:

$\sqrt{3}+i$

$\sqrt{3}+2i$

$2\sqrt{3}+2i$

$2\sqrt{3}+1$

$\sqrt{3}-1$

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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