Trigonometry Examples

$i^{22}$

Step 1

Rewrite $i^{22}$ as ${(i^{4})}^{5} i^{2}$ .

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Step 1.1

Factor out $i^{20}$ .

$i^{20} i^{2}$

Step 1.2

Rewrite $i^{20}$ as ${(i^{4})}^{5}$ .

${(i^{4})}^{5} i^{2}$

Step 2

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

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Step 2.1

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

${({(i^{2})}^{2})}^{5} i^{2}$

Step 2.2

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

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

Step 2.3

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

$1^{5} i^{2}$

Step 3

One to any power is one.

$1 i^{2}$

Step 4

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

$i^{2}$

Step 5

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

$- 1$

Step 6

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 7

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 8

Substitute the actual values of $a = - 1$ and $b = 0$ .

$| z | = \sqrt{0^{2} + {(- 1)}^{2}}$

Step 9

Find $| z |$ .

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Step 9.1

Raising $0$ to any positive power yields $0$ .

$| z | = \sqrt{0 + {(- 1)}^{2}}$

Step 9.2

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

$| z | = \sqrt{0 + 1}$

Step 9.3

Add $0$ and $1$ .

$| z | = \sqrt{1}$

Step 9.4

Any root of $1$ is $1$ .

$| z | = 1$

Step 10

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

$θ = \arctan (\frac{0}{- 1})$

Step 11

Since inverse tangent of $\frac{0}{- 1}$ produces an angle in the second quadrant, the value of the angle is $π$ .

$θ = π$

Step 12

Substitute the values of $θ = π$ and $| z | = 1$ .

$\cos (π) + i \sin (π)$