Trigonometry Examples

${(\sqrt{2})}^{4} (\cos (4 \cdot 120) + i \sin (4 \cdot 120))$

Step 1

Simplify by cancelling exponent with radical.

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Step 1.1

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

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Step 1.1.1

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

${(2^{\frac{1}{2}})}^{4} (\cos (4 \cdot 120) + i \sin (4 \cdot 120))$

Step 1.1.2

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

$2^{\frac{1}{2} \cdot 4} (\cos (4 \cdot 120) + i \sin (4 \cdot 120))$

Step 1.1.3

Combine $\frac{1}{2}$ and $4$ .

$2^{\frac{4}{2}} (\cos (4 \cdot 120) + i \sin (4 \cdot 120))$

Step 1.1.4

Cancel the common factor of $4$ and $2$ .

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Step 1.1.4.1

Factor $2$ out of $4$ .

$2^{\frac{2 \cdot 2}{2}} (\cos (4 \cdot 120) + i \sin (4 \cdot 120))$

Step 1.1.4.2

Cancel the common factors.

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Step 1.1.4.2.1

Factor $2$ out of $2$ .

$2^{\frac{2 \cdot 2}{2 (1)}} (\cos (4 \cdot 120) + i \sin (4 \cdot 120))$

Step 1.1.4.2.2

Cancel the common factor.

$2^{\frac{2 \cdot 2}{2 \cdot 1}} (\cos (4 \cdot 120) + i \sin (4 \cdot 120))$

Step 1.1.4.2.3

Rewrite the expression.

$2^{\frac{2}{1}} (\cos (4 \cdot 120) + i \sin (4 \cdot 120))$

Step 1.1.4.2.4

Divide $2$ by $1$ .

$2^{2} (\cos (4 \cdot 120) + i \sin (4 \cdot 120))$

Step 1.2

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

$4 (\cos (4 \cdot 120) + i \sin (4 \cdot 120))$

Step 2

Simplify each term.

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Step 2.1

Multiply $4$ by $120$ .

$4 (\cos (480) + i \sin (4 \cdot 120))$

Step 2.2

Remove full rotations of $360$ ° until the angle is between $0$ ° and $360$ °.

$4 (\cos (120) + i \sin (4 \cdot 120))$

Step 2.3

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.

$4 (- \cos (60) + i \sin (4 \cdot 120))$

Step 2.4

The exact value of $\cos (60)$ is $\frac{1}{2}$ .

$4 (- \frac{1}{2} + i \sin (4 \cdot 120))$

Step 2.5

Multiply $4$ by $120$ .

$4 (- \frac{1}{2} + i \sin (480))$

Step 2.6

Remove full rotations of $360$ ° until the angle is between $0$ ° and $360$ °.

$4 (- \frac{1}{2} + i \sin (120))$

Step 2.7

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

$4 (- \frac{1}{2} + i \sin (60))$

Step 2.8

The exact value of $\sin (60)$ is $\frac{\sqrt{3}}{2}$ .

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

Step 2.9

Combine $i$ and $\frac{\sqrt{3}}{2}$ .

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

Step 3

Simplify terms.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 3.2.2

Factor $2$ out of $4$ .

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

Step 3.2.3

Cancel the common factor.

$2 \cdot 2 \frac{- 1}{2} + 4 \frac{i \sqrt{3}}{2}$

Step 3.2.4

Rewrite the expression.

$2 \cdot - 1 + 4 \frac{i \sqrt{3}}{2}$

Step 3.3

Multiply $2$ by $- 1$ .

$- 2 + 4 \frac{i \sqrt{3}}{2}$

Step 3.4

Cancel the common factor of $2$ .

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Step 3.4.1

Factor $2$ out of $4$ .

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

Step 3.4.2

Cancel the common factor.

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

Step 3.4.3

Rewrite the expression.

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

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