Algebra Examples

$z = 2 (\cos (30 °) + i \sin (30 °))$

Step 1

Since $x = r \cos (θ)$ , replace $x$ with $r \cos (θ)$ .

$z = 2 (\cos (30 °) + i \sin (30 °))$

Step 2

Simplify the right side.

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Step 2.1

Simplify $2 (\cos (30 °) + i \sin (30 °))$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

The exact value of $\cos (30 °)$ is $\frac{\sqrt{3}}{2}$ .

$z = 2 (\frac{\sqrt{3}}{2} + i \sin (30 °))$

Step 2.1.1.2

The exact value of $\sin (30 °)$ is $\frac{1}{2}$ .

$z = 2 (\frac{\sqrt{3}}{2} + i \frac{1}{2})$

Step 2.1.1.3

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

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

Step 2.1.2

Simplify terms.

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Step 2.1.2.1

Apply the distributive property.

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

Step 2.1.2.2

Cancel the common factor of $2$ .

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Step 2.1.2.2.1

Cancel the common factor.

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

Step 2.1.2.2.2

Rewrite the expression.

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

Step 2.1.2.3

Cancel the common factor of $2$ .

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Step 2.1.2.3.1

Cancel the common factor.

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

Step 2.1.2.3.2

Rewrite the expression.

$z = \sqrt{3} + i$