Trigonometry Examples

${(1 + i)}^{8}$

Step 1

Use the Binomial Theorem.

$1^{8} + 8 \cdot 1^{7} i + 28 \cdot 1^{6} i^{2} + 56 \cdot 1^{5} i^{3} + 70 \cdot 1^{4} i^{4} + 56 \cdot 1^{3} i^{5} + 28 \cdot 1^{2} i^{6} + 8 \cdot 1 i^{7} + i^{8}$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

One to any power is one.

$1 + 8 \cdot 1^{7} i + 28 \cdot 1^{6} i^{2} + 56 \cdot 1^{5} i^{3} + 70 \cdot 1^{4} i^{4} + 56 \cdot 1^{3} i^{5} + 28 \cdot 1^{2} i^{6} + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.2

One to any power is one.

$1 + 8 \cdot 1 i + 28 \cdot 1^{6} i^{2} + 56 \cdot 1^{5} i^{3} + 70 \cdot 1^{4} i^{4} + 56 \cdot 1^{3} i^{5} + 28 \cdot 1^{2} i^{6} + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.3

Multiply $8$ by $1$ .

$1 + 8 i + 28 \cdot 1^{6} i^{2} + 56 \cdot 1^{5} i^{3} + 70 \cdot 1^{4} i^{4} + 56 \cdot 1^{3} i^{5} + 28 \cdot 1^{2} i^{6} + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.4

One to any power is one.

$1 + 8 i + 28 \cdot 1 i^{2} + 56 \cdot 1^{5} i^{3} + 70 \cdot 1^{4} i^{4} + 56 \cdot 1^{3} i^{5} + 28 \cdot 1^{2} i^{6} + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.5

Multiply $28$ by $1$ .

$1 + 8 i + 28 i^{2} + 56 \cdot 1^{5} i^{3} + 70 \cdot 1^{4} i^{4} + 56 \cdot 1^{3} i^{5} + 28 \cdot 1^{2} i^{6} + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.6

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

$1 + 8 i + 28 \cdot - 1 + 56 \cdot 1^{5} i^{3} + 70 \cdot 1^{4} i^{4} + 56 \cdot 1^{3} i^{5} + 28 \cdot 1^{2} i^{6} + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.7

Multiply $28$ by $- 1$ .

$1 + 8 i - 28 + 56 \cdot 1^{5} i^{3} + 70 \cdot 1^{4} i^{4} + 56 \cdot 1^{3} i^{5} + 28 \cdot 1^{2} i^{6} + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.8

One to any power is one.

$1 + 8 i - 28 + 56 \cdot 1 i^{3} + 70 \cdot 1^{4} i^{4} + 56 \cdot 1^{3} i^{5} + 28 \cdot 1^{2} i^{6} + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.9

Multiply $56$ by $1$ .

$1 + 8 i - 28 + 56 i^{3} + 70 \cdot 1^{4} i^{4} + 56 \cdot 1^{3} i^{5} + 28 \cdot 1^{2} i^{6} + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.10

Factor out $i^{2}$ .

$1 + 8 i - 28 + 56 (i^{2} \cdot i) + 70 \cdot 1^{4} i^{4} + 56 \cdot 1^{3} i^{5} + 28 \cdot 1^{2} i^{6} + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.11

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

$1 + 8 i - 28 + 56 (- 1 \cdot i) + 70 \cdot 1^{4} i^{4} + 56 \cdot 1^{3} i^{5} + 28 \cdot 1^{2} i^{6} + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.12

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

$1 + 8 i - 28 + 56 (- i) + 70 \cdot 1^{4} i^{4} + 56 \cdot 1^{3} i^{5} + 28 \cdot 1^{2} i^{6} + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.13

Multiply $- 1$ by $56$ .

$1 + 8 i - 28 - 56 i + 70 \cdot 1^{4} i^{4} + 56 \cdot 1^{3} i^{5} + 28 \cdot 1^{2} i^{6} + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.14

One to any power is one.

$1 + 8 i - 28 - 56 i + 70 \cdot 1 i^{4} + 56 \cdot 1^{3} i^{5} + 28 \cdot 1^{2} i^{6} + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.15

Multiply $70$ by $1$ .

$1 + 8 i - 28 - 56 i + 70 i^{4} + 56 \cdot 1^{3} i^{5} + 28 \cdot 1^{2} i^{6} + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.16

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

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Step 2.1.16.1

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

$1 + 8 i - 28 - 56 i + 70 {(i^{2})}^{2} + 56 \cdot 1^{3} i^{5} + 28 \cdot 1^{2} i^{6} + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.16.2

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

$1 + 8 i - 28 - 56 i + 70 {(- 1)}^{2} + 56 \cdot 1^{3} i^{5} + 28 \cdot 1^{2} i^{6} + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.16.3

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

$1 + 8 i - 28 - 56 i + 70 \cdot 1 + 56 \cdot 1^{3} i^{5} + 28 \cdot 1^{2} i^{6} + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.17

Multiply $70$ by $1$ .

$1 + 8 i - 28 - 56 i + 70 + 56 \cdot 1^{3} i^{5} + 28 \cdot 1^{2} i^{6} + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.18

One to any power is one.

$1 + 8 i - 28 - 56 i + 70 + 56 \cdot 1 i^{5} + 28 \cdot 1^{2} i^{6} + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.19

Multiply $56$ by $1$ .

$1 + 8 i - 28 - 56 i + 70 + 56 i^{5} + 28 \cdot 1^{2} i^{6} + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.20

Factor out $i^{4}$ .

$1 + 8 i - 28 - 56 i + 70 + 56 (i^{4} i) + 28 \cdot 1^{2} i^{6} + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.21

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

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Step 2.1.21.1

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

$1 + 8 i - 28 - 56 i + 70 + 56 ({(i^{2})}^{2} i) + 28 \cdot 1^{2} i^{6} + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.21.2

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

$1 + 8 i - 28 - 56 i + 70 + 56 ({(- 1)}^{2} i) + 28 \cdot 1^{2} i^{6} + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.21.3

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

$1 + 8 i - 28 - 56 i + 70 + 56 (1 i) + 28 \cdot 1^{2} i^{6} + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.22

Multiply $i$ by $1$ .

$1 + 8 i - 28 - 56 i + 70 + 56 i + 28 \cdot 1^{2} i^{6} + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.23

One to any power is one.

$1 + 8 i - 28 - 56 i + 70 + 56 i + 28 \cdot 1 i^{6} + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.24

Multiply $28$ by $1$ .

$1 + 8 i - 28 - 56 i + 70 + 56 i + 28 i^{6} + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.25

Factor out $i^{4}$ .

$1 + 8 i - 28 - 56 i + 70 + 56 i + 28 (i^{4} i^{2}) + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.26

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

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Step 2.1.26.1

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

$1 + 8 i - 28 - 56 i + 70 + 56 i + 28 ({(i^{2})}^{2} i^{2}) + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.26.2

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

$1 + 8 i - 28 - 56 i + 70 + 56 i + 28 ({(- 1)}^{2} i^{2}) + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.26.3

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

$1 + 8 i - 28 - 56 i + 70 + 56 i + 28 (1 i^{2}) + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.27

Multiply $i^{2}$ by $1$ .

$1 + 8 i - 28 - 56 i + 70 + 56 i + 28 i^{2} + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.28

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

$1 + 8 i - 28 - 56 i + 70 + 56 i + 28 \cdot - 1 + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.29

Multiply $28$ by $- 1$ .

$1 + 8 i - 28 - 56 i + 70 + 56 i - 28 + 8 \cdot 1 i^{7} + i^{8}$

Step 2.1.30

Multiply $8$ by $1$ .

$1 + 8 i - 28 - 56 i + 70 + 56 i - 28 + 8 i^{7} + i^{8}$

Step 2.1.31

Rewrite $i^{7}$ as $i^{4} (i^{2} \cdot i)$ .

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Step 2.1.31.1

Factor out $i^{4}$ .

$1 + 8 i - 28 - 56 i + 70 + 56 i - 28 + 8 (i^{4} i^{3}) + i^{8}$

Step 2.1.31.2

Factor out $i^{2}$ .

$1 + 8 i - 28 - 56 i + 70 + 56 i - 28 + 8 (i^{4} (i^{2} \cdot i)) + i^{8}$

Step 2.1.32

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

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Step 2.1.32.1

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

$1 + 8 i - 28 - 56 i + 70 + 56 i - 28 + 8 ({(i^{2})}^{2} (i^{2} \cdot i)) + i^{8}$

Step 2.1.32.2

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

$1 + 8 i - 28 - 56 i + 70 + 56 i - 28 + 8 ({(- 1)}^{2} (i^{2} \cdot i)) + i^{8}$

Step 2.1.32.3

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

$1 + 8 i - 28 - 56 i + 70 + 56 i - 28 + 8 (1 (i^{2} \cdot i)) + i^{8}$

Step 2.1.33

Multiply $i^{2} \cdot i$ by $1$ .

$1 + 8 i - 28 - 56 i + 70 + 56 i - 28 + 8 (i^{2} \cdot i) + i^{8}$

Step 2.1.34

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

$1 + 8 i - 28 - 56 i + 70 + 56 i - 28 + 8 (- 1 \cdot i) + i^{8}$

Step 2.1.35

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

$1 + 8 i - 28 - 56 i + 70 + 56 i - 28 + 8 (- i) + i^{8}$

Step 2.1.36

Multiply $- 1$ by $8$ .

$1 + 8 i - 28 - 56 i + 70 + 56 i - 28 - 8 i + i^{8}$

Step 2.1.37

Rewrite $i^{8}$ as ${(i^{4})}^{2}$ .

$1 + 8 i - 28 - 56 i + 70 + 56 i - 28 - 8 i + {(i^{4})}^{2}$

Step 2.1.38

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

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Step 2.1.38.1

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

$1 + 8 i - 28 - 56 i + 70 + 56 i - 28 - 8 i + {({(i^{2})}^{2})}^{2}$

Step 2.1.38.2

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

$1 + 8 i - 28 - 56 i + 70 + 56 i - 28 - 8 i + {({(- 1)}^{2})}^{2}$

Step 2.1.38.3

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

$1 + 8 i - 28 - 56 i + 70 + 56 i - 28 - 8 i + 1^{2}$

Step 2.1.39

One to any power is one.

$1 + 8 i - 28 - 56 i + 70 + 56 i - 28 - 8 i + 1$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $28$ from $1$ .

$- 27 + 8 i - 56 i + 70 + 56 i - 28 - 8 i + 1$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Add $- 27$ and $70$ .

$43 + 8 i - 56 i + 56 i - 28 - 8 i + 1$

Step 2.2.2.2

Subtract $28$ from $43$ .

$15 + 8 i - 56 i + 56 i - 8 i + 1$

Step 2.2.2.3

Add $15$ and $1$ .

$16 + 8 i - 56 i + 56 i - 8 i$

Step 2.2.3

Subtract $56 i$ from $8 i$ .

$16 - 48 i + 56 i - 8 i$

Step 2.2.4

Add $- 48 i$ and $56 i$ .

$16 + 8 i - 8 i$

Step 2.2.5

Subtract $8 i$ from $8 i$ .

$16 + 0$

Step 2.2.6

Add $16$ and $0$ .

$16$

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 $a = 16$ and $b = 0$ .

$| z | = \sqrt{0^{2} + 16^{2}}$

Step 6

Find $| z |$ .

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Step 6.1

Raising $0$ to any positive power yields $0$ .

$| z | = \sqrt{0 + 16^{2}}$

Step 6.2

Raise $16$ to the power of $2$ .

$| z | = \sqrt{0 + 256}$

Step 6.3

Add $0$ and $256$ .

$| z | = \sqrt{256}$

Step 6.4

Rewrite $256$ as $16^{2}$ .

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

Step 6.5

Pull terms out from under the radical, assuming positive real numbers.

$| z | = 16$

Step 7

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

$θ = \arctan (\frac{0}{16})$

Step 8

Since inverse tangent of $\frac{0}{16}$ produces an angle in the first quadrant, the value of the angle is $0$ .

$θ = 0$

Step 9

Substitute the values of $θ = 0$ and $| z | = 16$ .

$16 (\cos (0) + i \sin (0))$