Trigonometry Examples

${(1 - i)}^{7}$

Step 1

Use the Binomial Theorem.

$1^{7} + 7 \cdot 1^{6} (- i) + 21 \cdot 1^{5} {(- i)}^{2} + 35 \cdot 1^{4} {(- i)}^{3} + 35 \cdot 1^{3} {(- i)}^{4} + 21 \cdot 1^{2} {(- i)}^{5} + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

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.

$1 + 7 \cdot 1^{6} (- i) + 21 \cdot 1^{5} {(- i)}^{2} + 35 \cdot 1^{4} {(- i)}^{3} + 35 \cdot 1^{3} {(- i)}^{4} + 21 \cdot 1^{2} {(- i)}^{5} + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.2

One to any power is one.

$1 + 7 \cdot 1 (- i) + 21 \cdot 1^{5} {(- i)}^{2} + 35 \cdot 1^{4} {(- i)}^{3} + 35 \cdot 1^{3} {(- i)}^{4} + 21 \cdot 1^{2} {(- i)}^{5} + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.3

Multiply $7$ by $1$ .

$1 + 7 (- i) + 21 \cdot 1^{5} {(- i)}^{2} + 35 \cdot 1^{4} {(- i)}^{3} + 35 \cdot 1^{3} {(- i)}^{4} + 21 \cdot 1^{2} {(- i)}^{5} + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.4

Multiply $- 1$ by $7$ .

$1 - 7 i + 21 \cdot 1^{5} {(- i)}^{2} + 35 \cdot 1^{4} {(- i)}^{3} + 35 \cdot 1^{3} {(- i)}^{4} + 21 \cdot 1^{2} {(- i)}^{5} + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.5

One to any power is one.

$1 - 7 i + 21 \cdot 1 {(- i)}^{2} + 35 \cdot 1^{4} {(- i)}^{3} + 35 \cdot 1^{3} {(- i)}^{4} + 21 \cdot 1^{2} {(- i)}^{5} + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.6

Multiply $21$ by $1$ .

$1 - 7 i + 21 {(- i)}^{2} + 35 \cdot 1^{4} {(- i)}^{3} + 35 \cdot 1^{3} {(- i)}^{4} + 21 \cdot 1^{2} {(- i)}^{5} + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.7

Apply the product rule to $- i$ .

$1 - 7 i + 21 ({(- 1)}^{2} i^{2}) + 35 \cdot 1^{4} {(- i)}^{3} + 35 \cdot 1^{3} {(- i)}^{4} + 21 \cdot 1^{2} {(- i)}^{5} + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.8

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

$1 - 7 i + 21 (1 i^{2}) + 35 \cdot 1^{4} {(- i)}^{3} + 35 \cdot 1^{3} {(- i)}^{4} + 21 \cdot 1^{2} {(- i)}^{5} + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.9

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

$1 - 7 i + 21 i^{2} + 35 \cdot 1^{4} {(- i)}^{3} + 35 \cdot 1^{3} {(- i)}^{4} + 21 \cdot 1^{2} {(- i)}^{5} + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.10

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

$1 - 7 i + 21 \cdot - 1 + 35 \cdot 1^{4} {(- i)}^{3} + 35 \cdot 1^{3} {(- i)}^{4} + 21 \cdot 1^{2} {(- i)}^{5} + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.11

Multiply $21$ by $- 1$ .

$1 - 7 i - 21 + 35 \cdot 1^{4} {(- i)}^{3} + 35 \cdot 1^{3} {(- i)}^{4} + 21 \cdot 1^{2} {(- i)}^{5} + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.12

One to any power is one.

$1 - 7 i - 21 + 35 \cdot 1 {(- i)}^{3} + 35 \cdot 1^{3} {(- i)}^{4} + 21 \cdot 1^{2} {(- i)}^{5} + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.13

Multiply $35$ by $1$ .

$1 - 7 i - 21 + 35 {(- i)}^{3} + 35 \cdot 1^{3} {(- i)}^{4} + 21 \cdot 1^{2} {(- i)}^{5} + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.14

Apply the product rule to $- i$ .

$1 - 7 i - 21 + 35 ({(- 1)}^{3} i^{3}) + 35 \cdot 1^{3} {(- i)}^{4} + 21 \cdot 1^{2} {(- i)}^{5} + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.15

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

$1 - 7 i - 21 + 35 (- i^{3}) + 35 \cdot 1^{3} {(- i)}^{4} + 21 \cdot 1^{2} {(- i)}^{5} + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.16

Factor out $i^{2}$ .

$1 - 7 i - 21 + 35 (- (i^{2} \cdot i)) + 35 \cdot 1^{3} {(- i)}^{4} + 21 \cdot 1^{2} {(- i)}^{5} + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.17

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

$1 - 7 i - 21 + 35 (- (- 1 \cdot i)) + 35 \cdot 1^{3} {(- i)}^{4} + 21 \cdot 1^{2} {(- i)}^{5} + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.18

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

$1 - 7 i - 21 + 35 (- - i) + 35 \cdot 1^{3} {(- i)}^{4} + 21 \cdot 1^{2} {(- i)}^{5} + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.19

Multiply $- 1$ by $- 1$ .

$1 - 7 i - 21 + 35 (1 i) + 35 \cdot 1^{3} {(- i)}^{4} + 21 \cdot 1^{2} {(- i)}^{5} + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.20

Multiply $i$ by $1$ .

$1 - 7 i - 21 + 35 i + 35 \cdot 1^{3} {(- i)}^{4} + 21 \cdot 1^{2} {(- i)}^{5} + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.21

One to any power is one.

$1 - 7 i - 21 + 35 i + 35 \cdot 1 {(- i)}^{4} + 21 \cdot 1^{2} {(- i)}^{5} + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.22

Multiply $35$ by $1$ .

$1 - 7 i - 21 + 35 i + 35 {(- i)}^{4} + 21 \cdot 1^{2} {(- i)}^{5} + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.23

Apply the product rule to $- i$ .

$1 - 7 i - 21 + 35 i + 35 ({(- 1)}^{4} i^{4}) + 21 \cdot 1^{2} {(- i)}^{5} + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.24

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

$1 - 7 i - 21 + 35 i + 35 (1 i^{4}) + 21 \cdot 1^{2} {(- i)}^{5} + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.25

Multiply $i^{4}$ by $1$ .

$1 - 7 i - 21 + 35 i + 35 i^{4} + 21 \cdot 1^{2} {(- i)}^{5} + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.26

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

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Step 2.1.26.1

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

$1 - 7 i - 21 + 35 i + 35 {(i^{2})}^{2} + 21 \cdot 1^{2} {(- i)}^{5} + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.26.2

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

$1 - 7 i - 21 + 35 i + 35 {(- 1)}^{2} + 21 \cdot 1^{2} {(- i)}^{5} + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.26.3

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

$1 - 7 i - 21 + 35 i + 35 \cdot 1 + 21 \cdot 1^{2} {(- i)}^{5} + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.27

Multiply $35$ by $1$ .

$1 - 7 i - 21 + 35 i + 35 + 21 \cdot 1^{2} {(- i)}^{5} + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.28

One to any power is one.

$1 - 7 i - 21 + 35 i + 35 + 21 \cdot 1 {(- i)}^{5} + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.29

Multiply $21$ by $1$ .

$1 - 7 i - 21 + 35 i + 35 + 21 {(- i)}^{5} + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.30

Apply the product rule to $- i$ .

$1 - 7 i - 21 + 35 i + 35 + 21 ({(- 1)}^{5} i^{5}) + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.31

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

$1 - 7 i - 21 + 35 i + 35 + 21 (- i^{5}) + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.32

Factor out $i^{4}$ .

$1 - 7 i - 21 + 35 i + 35 + 21 (- (i^{4} i)) + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.33

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

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Step 2.1.33.1

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

$1 - 7 i - 21 + 35 i + 35 + 21 (- ({(i^{2})}^{2} i)) + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.33.2

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

$1 - 7 i - 21 + 35 i + 35 + 21 (- ({(- 1)}^{2} i)) + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.33.3

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

$1 - 7 i - 21 + 35 i + 35 + 21 (- (1 i)) + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.34

Multiply $i$ by $1$ .

$1 - 7 i - 21 + 35 i + 35 + 21 (- i) + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.35

Multiply $- 1$ by $21$ .

$1 - 7 i - 21 + 35 i + 35 - 21 i + 7 \cdot 1 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.36

Multiply $7$ by $1$ .

$1 - 7 i - 21 + 35 i + 35 - 21 i + 7 {(- i)}^{6} + {(- i)}^{7}$

Step 2.1.37

Apply the product rule to $- i$ .

$1 - 7 i - 21 + 35 i + 35 - 21 i + 7 ({(- 1)}^{6} i^{6}) + {(- i)}^{7}$

Step 2.1.38

Raise $- 1$ to the power of $6$ .

$1 - 7 i - 21 + 35 i + 35 - 21 i + 7 (1 i^{6}) + {(- i)}^{7}$

Step 2.1.39

Multiply $i^{6}$ by $1$ .

$1 - 7 i - 21 + 35 i + 35 - 21 i + 7 i^{6} + {(- i)}^{7}$

Step 2.1.40

Factor out $i^{4}$ .

$1 - 7 i - 21 + 35 i + 35 - 21 i + 7 (i^{4} i^{2}) + {(- i)}^{7}$

Step 2.1.41

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

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Step 2.1.41.1

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

$1 - 7 i - 21 + 35 i + 35 - 21 i + 7 ({(i^{2})}^{2} i^{2}) + {(- i)}^{7}$

Step 2.1.41.2

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

$1 - 7 i - 21 + 35 i + 35 - 21 i + 7 ({(- 1)}^{2} i^{2}) + {(- i)}^{7}$

Step 2.1.41.3

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

$1 - 7 i - 21 + 35 i + 35 - 21 i + 7 (1 i^{2}) + {(- i)}^{7}$

Step 2.1.42

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

$1 - 7 i - 21 + 35 i + 35 - 21 i + 7 i^{2} + {(- i)}^{7}$

Step 2.1.43

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

$1 - 7 i - 21 + 35 i + 35 - 21 i + 7 \cdot - 1 + {(- i)}^{7}$

Step 2.1.44

Multiply $7$ by $- 1$ .

$1 - 7 i - 21 + 35 i + 35 - 21 i - 7 + {(- i)}^{7}$

Step 2.1.45

Apply the product rule to $- i$ .

$1 - 7 i - 21 + 35 i + 35 - 21 i - 7 + {(- 1)}^{7} i^{7}$

Step 2.1.46

Raise $- 1$ to the power of $7$ .

$1 - 7 i - 21 + 35 i + 35 - 21 i - 7 - i^{7}$

Step 2.1.47

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

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Step 2.1.47.1

Factor out $i^{4}$ .

$1 - 7 i - 21 + 35 i + 35 - 21 i - 7 - (i^{4} i^{3})$

Step 2.1.47.2

Factor out $i^{2}$ .

$1 - 7 i - 21 + 35 i + 35 - 21 i - 7 - (i^{4} (i^{2} \cdot i))$

Step 2.1.48

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

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Step 2.1.48.1

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

$1 - 7 i - 21 + 35 i + 35 - 21 i - 7 - ({(i^{2})}^{2} (i^{2} \cdot i))$

Step 2.1.48.2

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

$1 - 7 i - 21 + 35 i + 35 - 21 i - 7 - ({(- 1)}^{2} (i^{2} \cdot i))$

Step 2.1.48.3

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

$1 - 7 i - 21 + 35 i + 35 - 21 i - 7 - (1 (i^{2} \cdot i))$

Step 2.1.49

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

$1 - 7 i - 21 + 35 i + 35 - 21 i - 7 - (i^{2} \cdot i)$

Step 2.1.50

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

$1 - 7 i - 21 + 35 i + 35 - 21 i - 7 - (- 1 \cdot i)$

Step 2.1.51

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

$1 - 7 i - 21 + 35 i + 35 - 21 i - 7 - - i$

Step 2.1.52

Multiply $- 1$ by $- 1$ .

$1 - 7 i - 21 + 35 i + 35 - 21 i - 7 + 1 i$

Step 2.1.53

Multiply $i$ by $1$ .

$1 - 7 i - 21 + 35 i + 35 - 21 i - 7 + i$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $21$ from $1$ .

$- 20 - 7 i + 35 i + 35 - 21 i - 7 + i$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Add $- 20$ and $35$ .

$15 - 7 i + 35 i - 21 i - 7 + i$

Step 2.2.2.2

Subtract $7$ from $15$ .

$8 - 7 i + 35 i - 21 i + i$

Step 2.2.3

Add $- 7 i$ and $35 i$ .

$8 + 28 i - 21 i + i$

Step 2.2.4

Subtract $21 i$ from $28 i$ .

$8 + 7 i + i$

Step 2.2.5

Add $7 i$ and $i$ .

$8 + 8 i$