Trigonometry Examples

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

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

One to any power is one.

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

Step 2.1.2

One to any power is one.

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

Step 2.1.3

Multiply $5$ by $1$ .

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

Step 2.1.4

Multiply $- 1$ by $5$ .

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

Step 2.1.5

One to any power is one.

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

Step 2.1.6

Multiply $10$ by $1$ .

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

Step 2.1.7

Apply the product rule to $- i$ .

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

Step 2.1.8

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

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

Step 2.1.9

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

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

Step 2.1.10

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

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

Step 2.1.11

Multiply $10$ by $- 1$ .

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

Step 2.1.12

One to any power is one.

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

Step 2.1.13

Multiply $10$ by $1$ .

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

Step 2.1.14

Apply the product rule to $- i$ .

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

Step 2.1.15

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

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

Step 2.1.16

Factor out $i^{2}$ .

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

Step 2.1.17

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

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

Step 2.1.18

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

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

Step 2.1.19

Multiply $- 1$ by $- 1$ .

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

Step 2.1.20

Multiply $i$ by $1$ .

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

Step 2.1.21

Multiply $5$ by $1$ .

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

Step 2.1.22

Apply the product rule to $- i$ .

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

Step 2.1.23

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

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

Step 2.1.24

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

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

Step 2.1.25

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

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Step 2.1.25.1

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

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

Step 2.1.25.2

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

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

Step 2.1.25.3

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

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

Step 2.1.26

Multiply $5$ by $1$ .

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

Step 2.1.27

Apply the product rule to $- i$ .

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

Step 2.1.28

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

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

Step 2.1.29

Factor out $i^{4}$ .

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

Step 2.1.30

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

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Step 2.1.30.1

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

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

Step 2.1.30.2

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

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

Step 2.1.30.3

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

$1 - 5 i - 10 + 10 i + 5 - (1 i)$

Step 2.1.31

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

Subtract $10$ from $1$ .

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

Step 2.2.2

Add $- 9$ and $5$ .

$- 4 - 5 i + 10 i - i$

Step 2.2.3

Add $- 5 i$ and $10 i$ .

$- 4 + 5 i - i$

Step 2.2.4

Subtract $i$ from $5 i$ .

$- 4 + 4 i$

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 second quadrant, the value of the angle is $\frac{3 π}{4}$ .

$θ = \frac{3 π}{4}$

Step 9

Substitute the values of $θ = \frac{3 π}{4}$ and $| z | = 4 \sqrt{2}$ .

$4 \sqrt{2} (\cos (\frac{3 π}{4}) + i \sin (\frac{3 π}{4}))$