Trigonometry Examples

${(- 1 - 2 i)}^{6}$

Step 1

Use the Binomial Theorem.

${(- 1)}^{6} + 6 {(- 1)}^{5} (- 2 i) + 15 {(- 1)}^{4} {(- 2 i)}^{2} + 20 {(- 1)}^{3} {(- 2 i)}^{3} + 15 {(- 1)}^{2} {(- 2 i)}^{4} + 6 \cdot - 1 {(- 2 i)}^{5} + {(- 2 i)}^{6}$

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 $6$ .

$1 + 6 {(- 1)}^{5} (- 2 i) + 15 {(- 1)}^{4} {(- 2 i)}^{2} + 20 {(- 1)}^{3} {(- 2 i)}^{3} + 15 {(- 1)}^{2} {(- 2 i)}^{4} + 6 \cdot - 1 {(- 2 i)}^{5} + {(- 2 i)}^{6}$

Step 2.1.2

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

$1 + 6 \cdot - 1 (- 2 i) + 15 {(- 1)}^{4} {(- 2 i)}^{2} + 20 {(- 1)}^{3} {(- 2 i)}^{3} + 15 {(- 1)}^{2} {(- 2 i)}^{4} + 6 \cdot - 1 {(- 2 i)}^{5} + {(- 2 i)}^{6}$

Step 2.1.3

Multiply $6$ by $- 1$ .

$1 - 6 (- 2 i) + 15 {(- 1)}^{4} {(- 2 i)}^{2} + 20 {(- 1)}^{3} {(- 2 i)}^{3} + 15 {(- 1)}^{2} {(- 2 i)}^{4} + 6 \cdot - 1 {(- 2 i)}^{5} + {(- 2 i)}^{6}$

Step 2.1.4

Multiply $- 2$ by $- 6$ .

$1 + 12 i + 15 {(- 1)}^{4} {(- 2 i)}^{2} + 20 {(- 1)}^{3} {(- 2 i)}^{3} + 15 {(- 1)}^{2} {(- 2 i)}^{4} + 6 \cdot - 1 {(- 2 i)}^{5} + {(- 2 i)}^{6}$

Step 2.1.5

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

$1 + 12 i + 15 \cdot 1 {(- 2 i)}^{2} + 20 {(- 1)}^{3} {(- 2 i)}^{3} + 15 {(- 1)}^{2} {(- 2 i)}^{4} + 6 \cdot - 1 {(- 2 i)}^{5} + {(- 2 i)}^{6}$

Step 2.1.6

Multiply $15$ by $1$ .

$1 + 12 i + 15 {(- 2 i)}^{2} + 20 {(- 1)}^{3} {(- 2 i)}^{3} + 15 {(- 1)}^{2} {(- 2 i)}^{4} + 6 \cdot - 1 {(- 2 i)}^{5} + {(- 2 i)}^{6}$

Step 2.1.7

Apply the product rule to $- 2 i$ .

$1 + 12 i + 15 ({(- 2)}^{2} i^{2}) + 20 {(- 1)}^{3} {(- 2 i)}^{3} + 15 {(- 1)}^{2} {(- 2 i)}^{4} + 6 \cdot - 1 {(- 2 i)}^{5} + {(- 2 i)}^{6}$

Step 2.1.8

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

$1 + 12 i + 15 (4 i^{2}) + 20 {(- 1)}^{3} {(- 2 i)}^{3} + 15 {(- 1)}^{2} {(- 2 i)}^{4} + 6 \cdot - 1 {(- 2 i)}^{5} + {(- 2 i)}^{6}$

Step 2.1.9

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

$1 + 12 i + 15 (4 \cdot - 1) + 20 {(- 1)}^{3} {(- 2 i)}^{3} + 15 {(- 1)}^{2} {(- 2 i)}^{4} + 6 \cdot - 1 {(- 2 i)}^{5} + {(- 2 i)}^{6}$

Step 2.1.10

Multiply $15 (4 \cdot - 1)$ .

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Step 2.1.10.1

Multiply $4$ by $- 1$ .

$1 + 12 i + 15 \cdot - 4 + 20 {(- 1)}^{3} {(- 2 i)}^{3} + 15 {(- 1)}^{2} {(- 2 i)}^{4} + 6 \cdot - 1 {(- 2 i)}^{5} + {(- 2 i)}^{6}$

Step 2.1.10.2

Multiply $15$ by $- 4$ .

$1 + 12 i - 60 + 20 {(- 1)}^{3} {(- 2 i)}^{3} + 15 {(- 1)}^{2} {(- 2 i)}^{4} + 6 \cdot - 1 {(- 2 i)}^{5} + {(- 2 i)}^{6}$

Step 2.1.11

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

$1 + 12 i - 60 + 20 \cdot - 1 {(- 2 i)}^{3} + 15 {(- 1)}^{2} {(- 2 i)}^{4} + 6 \cdot - 1 {(- 2 i)}^{5} + {(- 2 i)}^{6}$

Step 2.1.12

Multiply $20$ by $- 1$ .

$1 + 12 i - 60 - 20 {(- 2 i)}^{3} + 15 {(- 1)}^{2} {(- 2 i)}^{4} + 6 \cdot - 1 {(- 2 i)}^{5} + {(- 2 i)}^{6}$

Step 2.1.13

Apply the product rule to $- 2 i$ .

$1 + 12 i - 60 - 20 ({(- 2)}^{3} i^{3}) + 15 {(- 1)}^{2} {(- 2 i)}^{4} + 6 \cdot - 1 {(- 2 i)}^{5} + {(- 2 i)}^{6}$

Step 2.1.14

Raise $- 2$ to the power of $3$ .

$1 + 12 i - 60 - 20 (- 8 i^{3}) + 15 {(- 1)}^{2} {(- 2 i)}^{4} + 6 \cdot - 1 {(- 2 i)}^{5} + {(- 2 i)}^{6}$

Step 2.1.15

Factor out $i^{2}$ .

$1 + 12 i - 60 - 20 (- 8 (i^{2} \cdot i)) + 15 {(- 1)}^{2} {(- 2 i)}^{4} + 6 \cdot - 1 {(- 2 i)}^{5} + {(- 2 i)}^{6}$

Step 2.1.16

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

$1 + 12 i - 60 - 20 (- 8 (- 1 \cdot i)) + 15 {(- 1)}^{2} {(- 2 i)}^{4} + 6 \cdot - 1 {(- 2 i)}^{5} + {(- 2 i)}^{6}$

Step 2.1.17

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

$1 + 12 i - 60 - 20 (- 8 (- i)) + 15 {(- 1)}^{2} {(- 2 i)}^{4} + 6 \cdot - 1 {(- 2 i)}^{5} + {(- 2 i)}^{6}$

Step 2.1.18

Multiply $- 1$ by $- 8$ .

$1 + 12 i - 60 - 20 (8 i) + 15 {(- 1)}^{2} {(- 2 i)}^{4} + 6 \cdot - 1 {(- 2 i)}^{5} + {(- 2 i)}^{6}$

Step 2.1.19

Multiply $8$ by $- 20$ .

$1 + 12 i - 60 - 160 i + 15 {(- 1)}^{2} {(- 2 i)}^{4} + 6 \cdot - 1 {(- 2 i)}^{5} + {(- 2 i)}^{6}$

Step 2.1.20

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

$1 + 12 i - 60 - 160 i + 15 \cdot 1 {(- 2 i)}^{4} + 6 \cdot - 1 {(- 2 i)}^{5} + {(- 2 i)}^{6}$

Step 2.1.21

Multiply $15$ by $1$ .

$1 + 12 i - 60 - 160 i + 15 {(- 2 i)}^{4} + 6 \cdot - 1 {(- 2 i)}^{5} + {(- 2 i)}^{6}$

Step 2.1.22

Apply the product rule to $- 2 i$ .

$1 + 12 i - 60 - 160 i + 15 ({(- 2)}^{4} i^{4}) + 6 \cdot - 1 {(- 2 i)}^{5} + {(- 2 i)}^{6}$

Step 2.1.23

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

$1 + 12 i - 60 - 160 i + 15 (16 i^{4}) + 6 \cdot - 1 {(- 2 i)}^{5} + {(- 2 i)}^{6}$

Step 2.1.24

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

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Step 2.1.24.1

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

$1 + 12 i - 60 - 160 i + 15 (16 {(i^{2})}^{2}) + 6 \cdot - 1 {(- 2 i)}^{5} + {(- 2 i)}^{6}$

Step 2.1.24.2

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

$1 + 12 i - 60 - 160 i + 15 (16 {(- 1)}^{2}) + 6 \cdot - 1 {(- 2 i)}^{5} + {(- 2 i)}^{6}$

Step 2.1.24.3

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

$1 + 12 i - 60 - 160 i + 15 (16 \cdot 1) + 6 \cdot - 1 {(- 2 i)}^{5} + {(- 2 i)}^{6}$

Step 2.1.25

Multiply $15 (16 \cdot 1)$ .

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Step 2.1.25.1

Multiply $16$ by $1$ .

$1 + 12 i - 60 - 160 i + 15 \cdot 16 + 6 \cdot - 1 {(- 2 i)}^{5} + {(- 2 i)}^{6}$

Step 2.1.25.2

Multiply $15$ by $16$ .

$1 + 12 i - 60 - 160 i + 240 + 6 \cdot - 1 {(- 2 i)}^{5} + {(- 2 i)}^{6}$

Step 2.1.26

Multiply $6$ by $- 1$ .

$1 + 12 i - 60 - 160 i + 240 - 6 {(- 2 i)}^{5} + {(- 2 i)}^{6}$

Step 2.1.27

Apply the product rule to $- 2 i$ .

$1 + 12 i - 60 - 160 i + 240 - 6 ({(- 2)}^{5} i^{5}) + {(- 2 i)}^{6}$

Step 2.1.28

Raise $- 2$ to the power of $5$ .

$1 + 12 i - 60 - 160 i + 240 - 6 (- 32 i^{5}) + {(- 2 i)}^{6}$

Step 2.1.29

Factor out $i^{4}$ .

$1 + 12 i - 60 - 160 i + 240 - 6 (- 32 (i^{4} i)) + {(- 2 i)}^{6}$

Step 2.1.30

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

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Step 2.1.30.1

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

$1 + 12 i - 60 - 160 i + 240 - 6 (- 32 ({(i^{2})}^{2} i)) + {(- 2 i)}^{6}$

Step 2.1.30.2

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

$1 + 12 i - 60 - 160 i + 240 - 6 (- 32 ({(- 1)}^{2} i)) + {(- 2 i)}^{6}$

Step 2.1.30.3

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

$1 + 12 i - 60 - 160 i + 240 - 6 (- 32 (1 i)) + {(- 2 i)}^{6}$

Step 2.1.31

Multiply $i$ by $1$ .

$1 + 12 i - 60 - 160 i + 240 - 6 (- 32 i) + {(- 2 i)}^{6}$

Step 2.1.32

Multiply $- 32$ by $- 6$ .

$1 + 12 i - 60 - 160 i + 240 + 192 i + {(- 2 i)}^{6}$

Step 2.1.33

Apply the product rule to $- 2 i$ .

$1 + 12 i - 60 - 160 i + 240 + 192 i + {(- 2)}^{6} i^{6}$

Step 2.1.34

Raise $- 2$ to the power of $6$ .

$1 + 12 i - 60 - 160 i + 240 + 192 i + 64 i^{6}$

Step 2.1.35

Factor out $i^{4}$ .

$1 + 12 i - 60 - 160 i + 240 + 192 i + 64 (i^{4} i^{2})$

Step 2.1.36

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

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Step 2.1.36.1

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

$1 + 12 i - 60 - 160 i + 240 + 192 i + 64 ({(i^{2})}^{2} i^{2})$

Step 2.1.36.2

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

$1 + 12 i - 60 - 160 i + 240 + 192 i + 64 ({(- 1)}^{2} i^{2})$

Step 2.1.36.3

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

$1 + 12 i - 60 - 160 i + 240 + 192 i + 64 (1 i^{2})$

Step 2.1.37

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

$1 + 12 i - 60 - 160 i + 240 + 192 i + 64 i^{2}$

Step 2.1.38

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

$1 + 12 i - 60 - 160 i + 240 + 192 i + 64 \cdot - 1$

Step 2.1.39

Multiply $64$ by $- 1$ .

$1 + 12 i - 60 - 160 i + 240 + 192 i - 64$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $60$ from $1$ .

$- 59 + 12 i - 160 i + 240 + 192 i - 64$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Add $- 59$ and $240$ .

$181 + 12 i - 160 i + 192 i - 64$

Step 2.2.2.2

Subtract $64$ from $181$ .

$117 + 12 i - 160 i + 192 i$

Step 2.2.3

Subtract $160 i$ from $12 i$ .

$117 - 148 i + 192 i$

Step 2.2.4

Add $- 148 i$ and $192 i$ .

$117 + 44 i$

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 $a = 117$ and $b = 44$ .

$| z | = \sqrt{44^{2} + 117^{2}}$

Step 6

Find $| z |$ .

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Step 6.1

Raise $44$ to the power of $2$ .

$| z | = \sqrt{1936 + 117^{2}}$

Step 6.2

Raise $117$ to the power of $2$ .

$| z | = \sqrt{1936 + 13689}$

Step 6.3

Add $1936$ and $13689$ .

$| z | = \sqrt{15625}$

Step 6.4

Rewrite $15625$ as $125^{2}$ .

$| z | = \sqrt{125^{2}}$

Step 6.5

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

$| z | = 125$

Step 7

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

$θ = \arctan (\frac{44}{117})$

Step 8

Since inverse tangent of $\frac{44}{117}$ produces an angle in the first quadrant, the value of the angle is $0.35970699$ .

$θ = 0.35970699$

Step 9

Substitute the values of $θ = 0.35970699$ and $| z | = 125$ .

$125 (\cos (0.35970699) + i \sin (0.35970699))$