Algebra Examples

${(4 (\cos (105) + i \sin (105)))}^{3}$

Step 1

Simplify each term.

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Step 1.1

The exact value of $\cos (105)$ is $- \frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 1.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

${(4 (- \cos (75) + i \sin (105)))}^{3}$

Step 1.1.2

Split $75$ into two angles where the values of the six trigonometric functions are known.

${(4 (- \cos (30 + 45) + i \sin (105)))}^{3}$

Step 1.1.3

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

${(4 (- (\cos (30) \cos (45) - \sin (30) \sin (45)) + i \sin (105)))}^{3}$

Step 1.1.4

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

${(4 (- (\frac{\sqrt{3}}{2} \cos (45) - \sin (30) \sin (45)) + i \sin (105)))}^{3}$

Step 1.1.5

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

${(4 (- (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \sin (30) \sin (45)) + i \sin (105)))}^{3}$

Step 1.1.6

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

${(4 (- (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \sin (45)) + i \sin (105)))}^{3}$

Step 1.1.7

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

${(4 (- (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) + i \sin (105)))}^{3}$

Step 1.1.8

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

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Step 1.1.8.1

Simplify each term.

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Step 1.1.8.1.1

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

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Step 1.1.8.1.1.1

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

${(4 (- (\frac{\sqrt{3} \sqrt{2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) + i \sin (105)))}^{3}$

Step 1.1.8.1.1.2

Combine using the product rule for radicals.

${(4 (- (\frac{\sqrt{3 \cdot 2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) + i \sin (105)))}^{3}$

Step 1.1.8.1.1.3

Multiply $3$ by $2$ .

${(4 (- (\frac{\sqrt{6}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) + i \sin (105)))}^{3}$

Step 1.1.8.1.1.4

Multiply $2$ by $2$ .

${(4 (- (\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) + i \sin (105)))}^{3}$

Step 1.1.8.1.2

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

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Step 1.1.8.1.2.1

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

${(4 (- (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}) + i \sin (105)))}^{3}$

Step 1.1.8.1.2.2

Multiply $2$ by $2$ .

${(4 (- (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}) + i \sin (105)))}^{3}$

Step 1.1.8.2

Combine the numerators over the common denominator.

${(4 (- \frac{\sqrt{6} - \sqrt{2}}{4} + i \sin (105)))}^{3}$

Step 1.2

The exact value of $\sin (105)$ is $\frac{\sqrt{2} + \sqrt{6}}{4}$ .

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Step 1.2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

${(4 (- \frac{\sqrt{6} - \sqrt{2}}{4} + i \sin (75)))}^{3}$

Step 1.2.2

Split $75$ into two angles where the values of the six trigonometric functions are known.

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

Step 1.2.3

Apply the sum of angles identity.

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

Step 1.2.4

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

${(4 (- \frac{\sqrt{6} - \sqrt{2}}{4} + i (\frac{1}{2} \cos (45) + \cos (30) \sin (45))))}^{3}$

Step 1.2.5

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

${(4 (- \frac{\sqrt{6} - \sqrt{2}}{4} + i (\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \cos (30) \sin (45))))}^{3}$

Step 1.2.6

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

${(4 (- \frac{\sqrt{6} - \sqrt{2}}{4} + i (\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \sin (45))))}^{3}$

Step 1.2.7

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

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

Step 1.2.8

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

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Step 1.2.8.1

Simplify each term.

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Step 1.2.8.1.1

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

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Step 1.2.8.1.1.1

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

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

Step 1.2.8.1.1.2

Multiply $2$ by $2$ .

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

Step 1.2.8.1.2

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

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Step 1.2.8.1.2.1

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

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

Step 1.2.8.1.2.2

Combine using the product rule for radicals.

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

Step 1.2.8.1.2.3

Multiply $3$ by $2$ .

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

Step 1.2.8.1.2.4

Multiply $2$ by $2$ .

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

Step 1.2.8.2

Combine the numerators over the common denominator.

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

Step 1.3

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

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

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Combine the numerators over the common denominator.

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Step 2.1.1

Apply the distributive property.

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

Step 2.1.2

Multiply $- - \sqrt{2}$ .

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Step 2.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 2.1.2.2

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

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

Step 2.1.3

Apply the distributive property.

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

Step 2.2

Cancel the common factor of $4$ .

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

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