Convert to Rectangular (2(cos(15)+isin(15)))^3

${(2 (\cos (15) + i \sin (15)))}^{3}$

Step 1

Simplify each term.

The exact value of $\cos (15)$ is $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

Split $15$ into two angles where the values of the six trigonometric functions are known.

${(2 (\cos (45 - 30) + i \sin (15)))}^{3}$

Separate negation.

${(2 (\cos (45 - (30)) + i \sin (15)))}^{3}$

Apply the difference of angles identity $\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

${(2 (\cos (45) \cos (30) + \sin (45) \sin (30) + i \sin (15)))}^{3}$

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

${(2 (\frac{\sqrt{2}}{2} \cos (30) + \sin (45) \sin (30) + i \sin (15)))}^{3}$

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

${(2 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \sin (45) \sin (30) + i \sin (15)))}^{3}$

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

${(2 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \sin (30) + i \sin (15)))}^{3}$

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

${(2 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} + i \sin (15)))}^{3}$

Simplify $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

Simplify each term.

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}$ .

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

${(2 (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} + i \sin (15)))}^{3}$

Combine using the product rule for radicals.

${(2 (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} + i \sin (15)))}^{3}$

Multiply $2$ by $3$ .

${(2 (\frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} + i \sin (15)))}^{3}$

Multiply $2$ by $2$ .

${(2 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} + i \sin (15)))}^{3}$

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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

${(2 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2 \cdot 2} + i \sin (15)))}^{3}$

Multiply $2$ by $2$ .

${(2 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} + i \sin (15)))}^{3}$

Combine the numerators over the common denominator.

${(2 (\frac{\sqrt{6} + \sqrt{2}}{4} + i \sin (15)))}^{3}$

The exact value of $\sin (15)$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

Split $15$ into two angles where the values of the six trigonometric functions are known.

${(2 (\frac{\sqrt{6} + \sqrt{2}}{4} + i \sin (45 - 30)))}^{3}$

Separate negation.

${(2 (\frac{\sqrt{6} + \sqrt{2}}{4} + i \sin (45 - (30))))}^{3}$

Apply the difference of angles identity.

${(2 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\sin (45) \cos (30) - \cos (45) \sin (30))))}^{3}$

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

${(2 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{2}}{2} \cos (30) - \cos (45) \sin (30))))}^{3}$

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

${(2 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30))))}^{3}$

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

${(2 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (30))))}^{3}$

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

${(2 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})))}^{3}$

Simplify $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

Simplify each term.

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}$ .

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

${(2 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})))}^{3}$

Combine using the product rule for radicals.

${(2 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})))}^{3}$

Multiply $2$ by $3$ .

${(2 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})))}^{3}$

Multiply $2$ by $2$ .

${(2 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})))}^{3}$

Multiply $- \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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

${(2 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2})))}^{3}$

Multiply $2$ by $2$ .

${(2 (\frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4})))}^{3}$

Combine the numerators over the common denominator.

${(2 (\frac{\sqrt{6} + \sqrt{2}}{4} + i \frac{\sqrt{6} - \sqrt{2}}{4}))}^{3}$

Combine $i$ and $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

${(2 (\frac{\sqrt{6} + \sqrt{2}}{4} + \frac{i (\sqrt{6} - \sqrt{2})}{4}))}^{3}$

Step 2

Reduce the expression by cancelling the common factors.

Combine the numerators over the common denominator.

${(2 \frac{\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2}}{4})}^{3}$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

${(2 \frac{\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2}}{2 (2)})}^{3}$

Cancel the common factor.

${(2 \frac{\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2}}{2 \cdot 2})}^{3}$

Rewrite the expression.

${(\frac{\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2}}{2})}^{3}$

Simplify the expression.

Apply the product rule to $\frac{\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2}}{2}$ .

$\frac{{(\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}^{3}}{2^{3}}$

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

$\frac{{(\sqrt{6} + \sqrt{2} + i \sqrt{6} - i \sqrt{2})}^{3}}{8}$

Convert to Rectangular (2(cos(15)+isin(15)))^3

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