Trigonometry Examples

${(1 + i)}^{3}$

Step 1

Use the Binomial Theorem.

$1^{3} + 3 \cdot 1^{2} i + 3 \cdot 1 i^{2} + i^{3}$

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.

$1 + 3 \cdot 1^{2} i + 3 \cdot 1 i^{2} + i^{3}$

Step 2.1.2

One to any power is one.

$1 + 3 \cdot 1 i + 3 \cdot 1 i^{2} + i^{3}$

Step 2.1.3

Multiply $3$ by $1$ .

$1 + 3 i + 3 \cdot 1 i^{2} + i^{3}$

Step 2.1.4

Multiply $3$ by $1$ .

$1 + 3 i + 3 i^{2} + i^{3}$

Step 2.1.5

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

$1 + 3 i + 3 \cdot - 1 + i^{3}$

Step 2.1.6

Multiply $3$ by $- 1$ .

$1 + 3 i - 3 + i^{3}$

Step 2.1.7

Factor out $i^{2}$ .

$1 + 3 i - 3 + i^{2} \cdot i$

Step 2.1.8

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

$1 + 3 i - 3 - 1 \cdot i$

Step 2.1.9

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

$1 + 3 i - 3 - i$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $3$ from $1$ .

$- 2 + 3 i - i$

Step 2.2.2

Subtract $i$ from $3 i$ .

$- 2 + 2 i$

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 $a = - 2$ and $b = 2$ .

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

Step 6

Find $| z |$ .

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Step 6.1

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

$| z | = \sqrt{4 + {(- 2)}^{2}}$

Step 6.2

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

$| z | = \sqrt{4 + 4}$

Step 6.3

Add $4$ and $4$ .

$| z | = \sqrt{8}$

Step 6.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 6.4.1

Factor $4$ out of $8$ .

$| z | = \sqrt{4 (2)}$

Step 6.4.2

Rewrite $4$ as $2^{2}$ .

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

Step 6.5

Pull terms out from under the radical.

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

Step 7

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

$θ = \arctan (\frac{2}{- 2})$

Step 8

Since inverse tangent of $\frac{2}{- 2}$ 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 $| z | = 2 \sqrt{2}$ .

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