Convert to Trigonometric Form i^22

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Trigonometry

$i^{22}$

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

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Factor out $i^{20}$ .

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

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

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

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

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

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

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

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

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

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

One to any power is one.

$1 i^{2}$

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

$i^{2}$

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

$- 1$

This is the trigonometric form of a complex number where $| z |$ is the modulus and $θ$ is the angle created on the complex plane.

$z = a + b i = | z | (\cos (θ) + i \sin (θ))$

The modulus of a complex number is the distance from the origin on the complex plane.

$| z | = \sqrt{a^{2} + b^{2}}$ where $z = a + b i$

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

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

Find $| z |$ .

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

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

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

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

Add $0$ and $1$ .

$| z | = \sqrt{1}$

Any root of $1$ is $1$ .

$| z | = 1$

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

$θ = \arctan (\frac{0}{- 1})$

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

$θ = π$

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

$\cos (π) + i \sin (π)$

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Name

Name	one billion five hundred eighty-six million six hundred seventy-four thousand seven hundred forty-three

Interesting facts

1586674743 has 8 divisors, whose sum is 2226912000
The reverse of 1586674743 is 3474766851
Previous prime number is 19

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 51
Digital Root 6

Name

Name	four hundred twenty-eight million two hundred thirty-two thousand seven hundred ninety-two

Interesting facts

428232792 has 256 divisors, whose sum is 2189799360
The reverse of 428232792 is 297232824
Previous prime number is 13

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 39
Digital Root 3

Name

Name	one billion five hundred eighty-one million six hundred fourteen thousand six hundred eight

Interesting facts

1581614608 has 128 divisors, whose sum is 9164803968
The reverse of 1581614608 is 8064161851
Previous prime number is 673

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 40
Digital Root 4

Solve for x 50=25sin((2pi)/7t)+75

Solve for ? x=(2pi)/3

Verify the Identity 1/(1+cos(x))-1/(1-cos(x))=-2csc(x)cot(x)

Graph y=- square root of x

Graph y=cos(5x)