Convert to Trigonometric Form square root of 3+3i

$\sqrt{3} + 3 i$

This is the trigonometric form of a complex number where $| z |$ is the modulus and $θ$ is the angle created on the complex plane.

$z = a + b i = | z | (\cos (θ) + i \sin (θ))$

The modulus of a complex number is the distance from the origin on the complex plane.

$| z | = \sqrt{a^{2} + b^{2}}$ where $z = a + b i$

Substitute the actual values of $a = \sqrt{3}$ and $b = 3$ .

$| z | = \sqrt{3^{2} + {(\sqrt{3})}^{2}}$

Find $| z |$ .

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Raise $3$ to the power of $2$ .

$| z | = \sqrt{9 + {(\sqrt{3})}^{2}}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$| z | = \sqrt{9 + {(3^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$| z | = \sqrt{9 + 3^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$| z | = \sqrt{9 + 3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$| z | = \sqrt{9 + 3^{\frac{2}{2}}}$

Rewrite the expression.

$| z | = \sqrt{9 + 3}$

Evaluate the exponent.

$| z | = \sqrt{9 + 3}$

Add $9$ and $3$ .

$| z | = \sqrt{12}$

Rewrite $12$ as $2^{2} \cdot 3$ .

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Factor $4$ out of $12$ .

$| z | = \sqrt{4 (3)}$

Rewrite $4$ as $2^{2}$ .

$| z | = \sqrt{2^{2} \cdot 3}$

Pull terms out from under the radical.

$| z | = 2 \sqrt{3}$

The angle of the point on the complex plane is the inverse tangent of the complex portion over the real portion.

$θ = \arctan (\frac{3}{\sqrt{3}})$

Since inverse tangent of $\frac{3}{\sqrt{3}}$ produces an angle in the first quadrant, the value of the angle is $\frac{π}{3}$ .

$θ = \frac{π}{3}$

Substitute the values of $θ = \frac{π}{3}$ and $| z | = 2 \sqrt{3}$ .

$2 \sqrt{3} (\cos (\frac{π}{3}) + i \sin (\frac{π}{3}))$

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Name

Name	one billion six hundred ninety million one hundred thirty-eight thousand eight hundred ninety-nine

Interesting facts

1690138899 has 8 divisors, whose sum is 1753481520
The reverse of 1690138899 is 9988310961
Previous prime number is 23993

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 54
Digital Root 9

Name

Name	one billion two hundred eighty million three hundred ninety-seven thousand nine hundred twenty-eight

Interesting facts

1280397928 has 32 divisors, whose sum is 4421840544
The reverse of 1280397928 is 8297930821
Previous prime number is 43

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 49
Digital Root 4

Name

Name	three hundred twenty million nine hundred eighty-two thousand fifty-one

Interesting facts

320982051 has 4 divisors, whose sum is 329212400
The reverse of 320982051 is 150289023
Previous prime number is 39

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 30
Digital Root 3

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Solve for x a=38cot(30)