Trigonometry Examples

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

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

Step 1

Simplify by cancelling exponent with radical.

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Step 1.1

Rewrite

{\sqrt{2}}^{4}

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

2^{2}

$2^{2}$ .

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Step 1.1.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{2}

$\sqrt{2}$ as

2^{\frac{1}{2}}

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

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

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

Step 1.1.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 1.1.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

4

$4$ .

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

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

Step 1.1.4

Cancel the common factor of

4

$4$ and

2

$2$ .

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Step 1.1.4.1

Factor

2

$2$ out of

4

$4$ .

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

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

Step 1.1.4.2

Cancel the common factors.

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Step 1.1.4.2.1

Factor

2

$2$ out of

2

$2$ .

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

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

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))

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

Step 1.1.4.2.3

Rewrite the expression.

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

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

Step 1.1.4.2.4

Divide

2

$2$ by

1

$1$ .

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

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

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

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

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

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

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

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

Step 1.2

Raise

2

$2$ to the power of

2

$2$ .

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

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

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

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

Step 2

Simplify each term.

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Step 2.1

Multiply

4

$4$ by

120

$120$ .

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

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

Step 2.2

Remove full rotations of

360

$360$ ° until the angle is between

0

$0$ ° and

360

$360$ °.

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

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

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))

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

Step 2.4

The exact value of

\cos (60)

$\cos (60)$ is

\frac{1}{2}

$\frac{1}{2}$ .

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

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

Step 2.5

Multiply

4

$4$ by

120

$120$ .

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

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

Step 2.6

Remove full rotations of

360

$360$ ° until the angle is between

0

$0$ ° and

360

$360$ °.

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

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

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))

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

Step 2.8

The exact value of

\sin (60)

$\sin (60)$ is

\frac{\sqrt{3}}{2}

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

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

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

Step 2.9

Combine

i

$i$ and

\frac{\sqrt{3}}{2}

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

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

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

4 (- \frac{1}{2} + \frac{i \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}

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

Step 3.2

Cancel the common factor of

2

$2$ .

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Step 3.2.1

Move the leading negative in

- \frac{1}{2}

$- \frac{1}{2}$ into the numerator.

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

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

Step 3.2.2

Factor

2

$2$ out of

4

$4$ .

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

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

Step 3.2.3

Cancel the common factor.

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

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

Step 3.2.4

Rewrite the expression.

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

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

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

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

Step 3.3

Multiply

2

$2$ by

- 1

$- 1$ .

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

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

Step 3.4

Cancel the common factor of

2

$2$ .

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Step 3.4.1

Factor

2

$2$ out of

4

$4$ .

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

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

Step 3.4.2

Cancel the common factor.

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

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

Step 3.4.3

Rewrite the expression.

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

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

- 2 + 2 i \sqrt{3}

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

- 2 + 2 i \sqrt{3}

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