Trigonometry Examples

${(- 1 + i)}^{5}$

Step 1

Use the Binomial Theorem.

${(- 1)}^{5} + 5 {(- 1)}^{4} i + 10 {(- 1)}^{3} i^{2} + 10 {(- 1)}^{2} i^{3} + 5 \cdot - 1 i^{4} + i^{5}$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Raise $- 1$ to the power of $5$ .

$- 1 + 5 {(- 1)}^{4} i + 10 {(- 1)}^{3} i^{2} + 10 {(- 1)}^{2} i^{3} + 5 \cdot - 1 i^{4} + i^{5}$

Step 2.1.2

Raise $- 1$ to the power of $4$ .

$- 1 + 5 \cdot 1 i + 10 {(- 1)}^{3} i^{2} + 10 {(- 1)}^{2} i^{3} + 5 \cdot - 1 i^{4} + i^{5}$

Step 2.1.3

Multiply $5$ by $1$ .

$- 1 + 5 i + 10 {(- 1)}^{3} i^{2} + 10 {(- 1)}^{2} i^{3} + 5 \cdot - 1 i^{4} + i^{5}$

Step 2.1.4

Raise $- 1$ to the power of $3$ .

$- 1 + 5 i + 10 \cdot - 1 i^{2} + 10 {(- 1)}^{2} i^{3} + 5 \cdot - 1 i^{4} + i^{5}$

Step 2.1.5

Multiply $10$ by $- 1$ .

$- 1 + 5 i - 10 i^{2} + 10 {(- 1)}^{2} i^{3} + 5 \cdot - 1 i^{4} + i^{5}$

Step 2.1.6

Rewrite $i^{2}$ as $- 1$ .

$- 1 + 5 i - 10 \cdot - 1 + 10 {(- 1)}^{2} i^{3} + 5 \cdot - 1 i^{4} + i^{5}$

Step 2.1.7

Multiply $- 10$ by $- 1$ .

$- 1 + 5 i + 10 + 10 {(- 1)}^{2} i^{3} + 5 \cdot - 1 i^{4} + i^{5}$

Step 2.1.8

Raise $- 1$ to the power of $2$ .

$- 1 + 5 i + 10 + 10 \cdot 1 i^{3} + 5 \cdot - 1 i^{4} + i^{5}$

Step 2.1.9

Multiply $10$ by $1$ .

$- 1 + 5 i + 10 + 10 i^{3} + 5 \cdot - 1 i^{4} + i^{5}$

Step 2.1.10

Factor out $i^{2}$ .

$- 1 + 5 i + 10 + 10 (i^{2} \cdot i) + 5 \cdot - 1 i^{4} + i^{5}$

Step 2.1.11

Rewrite $i^{2}$ as $- 1$ .

$- 1 + 5 i + 10 + 10 (- 1 \cdot i) + 5 \cdot - 1 i^{4} + i^{5}$

Step 2.1.12

Rewrite $- 1 i$ as $- i$ .

$- 1 + 5 i + 10 + 10 (- i) + 5 \cdot - 1 i^{4} + i^{5}$

Step 2.1.13

Multiply $- 1$ by $10$ .

$- 1 + 5 i + 10 - 10 i + 5 \cdot - 1 i^{4} + i^{5}$

Step 2.1.14

Multiply $5$ by $- 1$ .

$- 1 + 5 i + 10 - 10 i - 5 i^{4} + i^{5}$

Step 2.1.15

Rewrite $i^{4}$ as $1$ .

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Step 2.1.15.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$- 1 + 5 i + 10 - 10 i - 5 {(i^{2})}^{2} + i^{5}$

Step 2.1.15.2

Rewrite $i^{2}$ as $- 1$ .

$- 1 + 5 i + 10 - 10 i - 5 {(- 1)}^{2} + i^{5}$

Step 2.1.15.3

Raise $- 1$ to the power of $2$ .

$- 1 + 5 i + 10 - 10 i - 5 \cdot 1 + i^{5}$

Step 2.1.16

Multiply $- 5$ by $1$ .

$- 1 + 5 i + 10 - 10 i - 5 + i^{5}$

Step 2.1.17

Factor out $i^{4}$ .

$- 1 + 5 i + 10 - 10 i - 5 + i^{4} i$

Step 2.1.18

Rewrite $i^{4}$ as $1$ .

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Step 2.1.18.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$- 1 + 5 i + 10 - 10 i - 5 + {(i^{2})}^{2} i$

Step 2.1.18.2

Rewrite $i^{2}$ as $- 1$ .

$- 1 + 5 i + 10 - 10 i - 5 + {(- 1)}^{2} i$

Step 2.1.18.3

Raise $- 1$ to the power of $2$ .

$- 1 + 5 i + 10 - 10 i - 5 + 1 i$

Step 2.1.19

Multiply $i$ by $1$ .

$- 1 + 5 i + 10 - 10 i - 5 + i$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Add $- 1$ and $10$ .

$9 + 5 i - 10 i - 5 + i$

Step 2.2.2

Subtract $5$ from $9$ .

$4 + 5 i - 10 i + i$

Step 2.2.3

Subtract $10 i$ from $5 i$ .

$4 - 5 i + i$

Step 2.2.4

Add $- 5 i$ and $i$ .

$4 - 4 i$

Step 3

This is the trigonometric form of a complex number where $| z |$ is the modulus and $θ$ is the angle created on the complex plane.

$z = a + b i = | z | (\cos (θ) + i \sin (θ))$

Step 4

The modulus of a complex number is the distance from the origin on the complex plane.

$| z | = \sqrt{a^{2} + b^{2}}$ where $z = a + b i$

Step 5

Substitute the actual values of $a = 4$ and $b = - 4$ .

$| z | = \sqrt{{(- 4)}^{2} + 4^{2}}$

Step 6

Find $| z |$ .

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Step 6.1

Raise $- 4$ to the power of $2$ .

$| z | = \sqrt{16 + 4^{2}}$

Step 6.2

Raise $4$ to the power of $2$ .

$| z | = \sqrt{16 + 16}$

Step 6.3

Add $16$ and $16$ .

$| z | = \sqrt{32}$

Step 6.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 6.4.1

Factor $16$ out of $32$ .

$| z | = \sqrt{16 (2)}$

Step 6.4.2

Rewrite $16$ as $4^{2}$ .

$| z | = \sqrt{4^{2} \cdot 2}$

Step 6.5

Pull terms out from under the radical.

$| z | = 4 \sqrt{2}$

Step 7

The angle of the point on the complex plane is the inverse tangent of the complex portion over the real portion.

$θ = \arctan (\frac{- 4}{4})$

Step 8

Since inverse tangent of $\frac{- 4}{4}$ produces an angle in the fourth quadrant, the value of the angle is $- \frac{π}{4}$ .

$θ = - \frac{π}{4}$

Step 9

Substitute the values of $θ = - \frac{π}{4}$ and $| z | = 4 \sqrt{2}$ .

$4 \sqrt{2} (\cos (- \frac{π}{4}) + i \sin (- \frac{π}{4}))$