Algebra Examples

$- \frac{\sqrt{3}}{2} - \frac{1}{2} i$

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Substitute the actual values of $a = - \frac{\sqrt{3}}{2}$ and $b = - \frac{1}{2}$ .

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

Step 5

Find $| z |$ .

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Step 5.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.1.1

Apply the product rule to $- \frac{1}{2}$ .

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

Step 5.1.2

Apply the product rule to $\frac{1}{2}$ .

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

Step 5.2

Raise $- 1$ to the power of $2$ .

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

Step 5.3

Multiply $\frac{1^{2}}{2^{2}}$ by $1$ .

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

Step 5.4

One to any power is one.

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

Step 5.5

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

$| z | = \sqrt{\frac{1}{4} + {(- \frac{\sqrt{3}}{2})}^{2}}$

Step 5.6

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.6.1

Apply the product rule to $- \frac{\sqrt{3}}{2}$ .

$| z | = \sqrt{\frac{1}{4} + {(- 1)}^{2} {(\frac{\sqrt{3}}{2})}^{2}}$

Step 5.6.2

Apply the product rule to $\frac{\sqrt{3}}{2}$ .

$| z | = \sqrt{\frac{1}{4} + {(- 1)}^{2} (\frac{{\sqrt{3}}^{2}}{2^{2}})}$

Step 5.7

Simplify the expression.

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Step 5.7.1

Raise $- 1$ to the power of $2$ .

$| z | = \sqrt{\frac{1}{4} + 1 (\frac{{\sqrt{3}}^{2}}{2^{2}})}$

Step 5.7.2

Multiply $\frac{{\sqrt{3}}^{2}}{2^{2}}$ by $1$ .

$| z | = \sqrt{\frac{1}{4} + \frac{{\sqrt{3}}^{2}}{2^{2}}}$

Step 5.8

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

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Step 5.8.1

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

$| z | = \sqrt{\frac{1}{4} + \frac{{(3^{\frac{1}{2}})}^{2}}{2^{2}}}$

Step 5.8.2

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

$| z | = \sqrt{\frac{1}{4} + \frac{3^{\frac{1}{2} \cdot 2}}{2^{2}}}$

Step 5.8.3

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

$| z | = \sqrt{\frac{1}{4} + \frac{3^{\frac{2}{2}}}{2^{2}}}$

Step 5.8.4

Cancel the common factor of $2$ .

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Step 5.8.4.1

Cancel the common factor.

$| z | = \sqrt{\frac{1}{4} + \frac{3^{\frac{2}{2}}}{2^{2}}}$

Step 5.8.4.2

Rewrite the expression.

$| z | = \sqrt{\frac{1}{4} + \frac{3}{2^{2}}}$

Step 5.8.5

Evaluate the exponent.

$| z | = \sqrt{\frac{1}{4} + \frac{3}{2^{2}}}$

Step 5.9

Simplify the expression.

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Step 5.9.1

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

$| z | = \sqrt{\frac{1}{4} + \frac{3}{4}}$

Step 5.9.2

Combine the numerators over the common denominator.

$| z | = \sqrt{\frac{1 + 3}{4}}$

Step 5.9.3

Add $1$ and $3$ .

$| z | = \sqrt{\frac{4}{4}}$

Step 5.9.4

Divide $4$ by $4$ .

$| z | = \sqrt{1}$

Step 5.9.5

Any root of $1$ is $1$ .

$| z | = 1$

Step 6

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

$θ = \arctan (\frac{- \frac{1}{2}}{- \frac{\sqrt{3}}{2}})$

Step 7

Since inverse tangent of $\frac{- \frac{1}{2}}{- \frac{\sqrt{3}}{2}}$ produces an angle in the third quadrant, the value of the angle is $\frac{7 π}{6}$ .

$θ = \frac{7 π}{6}$

Step 8

Substitute the values of $θ = \frac{7 π}{6}$ and $| z | = 1$ .

$\cos (\frac{7 π}{6}) + i \sin (\frac{7 π}{6})$