Trigonometry Examples

${(1 + i)}^{6}$

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify terms.

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

One to any power is one.

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

Step 2.1.2

One to any power is one.

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

Step 2.1.3

Multiply $6$ by $1$ .

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

Step 2.1.4

One to any power is one.

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

Step 2.1.5

Multiply $15$ by $1$ .

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

Step 2.1.6

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

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

Step 2.1.7

Multiply $15$ by $- 1$ .

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

Step 2.1.8

One to any power is one.

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

Step 2.1.9

Multiply $20$ by $1$ .

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

Step 2.1.10

Factor out $i^{2}$ .

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

Step 2.1.11

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

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

Step 2.1.12

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

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

Step 2.1.13

Multiply $- 1$ by $20$ .

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

Step 2.1.14

One to any power is one.

$1 + 6 i - 15 - 20 i + 15 \cdot 1 i^{4} + 6 \cdot 1 i^{5} + i^{6}$

Step 2.1.15

Multiply $15$ by $1$ .

$1 + 6 i - 15 - 20 i + 15 i^{4} + 6 \cdot 1 i^{5} + i^{6}$

Step 2.1.16

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

Tap for more steps...

Step 2.1.16.1

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

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

Step 2.1.16.2

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

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

Step 2.1.16.3

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

$1 + 6 i - 15 - 20 i + 15 \cdot 1 + 6 \cdot 1 i^{5} + i^{6}$

Step 2.1.17

Multiply $15$ by $1$ .

$1 + 6 i - 15 - 20 i + 15 + 6 \cdot 1 i^{5} + i^{6}$

Step 2.1.18

Multiply $6$ by $1$ .

$1 + 6 i - 15 - 20 i + 15 + 6 i^{5} + i^{6}$

Step 2.1.19

Factor out $i^{4}$ .

$1 + 6 i - 15 - 20 i + 15 + 6 (i^{4} i) + i^{6}$

Step 2.1.20

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

Tap for more steps...

Step 2.1.20.1

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

$1 + 6 i - 15 - 20 i + 15 + 6 ({(i^{2})}^{2} i) + i^{6}$

Step 2.1.20.2

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

$1 + 6 i - 15 - 20 i + 15 + 6 ({(- 1)}^{2} i) + i^{6}$

Step 2.1.20.3

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

$1 + 6 i - 15 - 20 i + 15 + 6 (1 i) + i^{6}$

Step 2.1.21

Multiply $i$ by $1$ .

$1 + 6 i - 15 - 20 i + 15 + 6 i + i^{6}$

Step 2.1.22

Factor out $i^{4}$ .

$1 + 6 i - 15 - 20 i + 15 + 6 i + i^{4} i^{2}$

Step 2.1.23

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

Tap for more steps...

Step 2.1.23.1

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

$1 + 6 i - 15 - 20 i + 15 + 6 i + {(i^{2})}^{2} i^{2}$

Step 2.1.23.2

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

$1 + 6 i - 15 - 20 i + 15 + 6 i + {(- 1)}^{2} i^{2}$

Step 2.1.23.3

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

$1 + 6 i - 15 - 20 i + 15 + 6 i + 1 i^{2}$

Step 2.1.24

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

$1 + 6 i - 15 - 20 i + 15 + 6 i + i^{2}$

Step 2.1.25

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

$1 + 6 i - 15 - 20 i + 15 + 6 i - 1$

Step 2.2

Simplify by adding terms.

Tap for more steps...

Step 2.2.1

Subtract $15$ from $1$ .

$- 14 + 6 i - 20 i + 15 + 6 i - 1$

Step 2.2.2

Simplify by adding and subtracting.

Tap for more steps...

Step 2.2.2.1

Add $- 14$ and $15$ .

$1 + 6 i - 20 i + 6 i - 1$

Step 2.2.2.2

Subtract $1$ from $1$ .

$0 + 6 i - 20 i + 6 i$

Step 2.2.2.3

Add $0$ and $6 i$ .

$6 i - 20 i + 6 i$

Step 2.2.3

Subtract $20 i$ from $6 i$ .

$- 14 i + 6 i$

Step 2.2.4

Add $- 14 i$ and $6 i$ .

$- 8 i$

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 $a = 0$ and $b = - 8$ .

$| z | = \sqrt{{(- 8)}^{2}}$

Step 6

Find $| z |$ .

Tap for more steps...

Step 6.1

Raise $- 8$ to the power of $2$ .

$| z | = \sqrt{64}$

Step 6.2

Rewrite $64$ as $8^{2}$ .

$| z | = \sqrt{8^{2}}$

Step 6.3

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

$| z | = 8$

Step 7

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

$θ = \arctan (\frac{- 8}{0})$

Step 8

Since the argument is undefined and $b$ is negative, the angle of the point on the complex plane is $\frac{3 π}{2}$ .

$θ = \frac{3 π}{2}$

Step 9

Substitute the values of $θ = \frac{3 π}{2}$ and $| z | = 8$ .

$8 (\cos (\frac{3 π}{2}) + i \sin (\frac{3 π}{2}))$