Trigonometry Examples

$i^{- 15}$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{i^{15}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $i^{15}$ as ${(i^{4})}^{3} (i^{2} \cdot i)$ .

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Step 2.1.1

Factor out $i^{12}$ .

$\frac{1}{i^{12} i^{3}}$

Step 2.1.2

Rewrite $i^{12}$ as ${(i^{4})}^{3}$ .

$\frac{1}{{(i^{4})}^{3} i^{3}}$

Step 2.1.3

Factor out $i^{2}$ .

$\frac{1}{{(i^{4})}^{3} (i^{2} \cdot i)}$

Step 2.2

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

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Step 2.2.1

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

$\frac{1}{{({(i^{2})}^{2})}^{3} (i^{2} \cdot i)}$

Step 2.2.2

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

$\frac{1}{{({(- 1)}^{2})}^{3} (i^{2} \cdot i)}$

Step 2.2.3

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

$\frac{1}{1^{3} (i^{2} \cdot i)}$

Step 2.3

One to any power is one.

$\frac{1}{1 (i^{2} \cdot i)}$

Step 2.4

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

$\frac{1}{i^{2} \cdot i}$

Step 2.5

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

$\frac{1}{- 1 \cdot i}$

Step 2.6

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

$\frac{1}{- i}$

Step 3

Cancel the common factor of $1$ and $- 1$ .

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Step 3.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{- i}$

Step 3.2

Move the negative in front of the fraction.

$- \frac{1}{i}$

Step 4

Multiply the numerator and denominator of $\frac{1}{i}$ by the conjugate of $i$ to make the denominator real.

$- (\frac{1}{i} \cdot \frac{i}{i})$

Step 5

Multiply.

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Step 5.1

Combine.

$- \frac{1 i}{i i}$

Step 5.2

Multiply $i$ by $1$ .

$- \frac{i}{i i}$

Step 5.3

Simplify the denominator.

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Step 5.3.1

Raise $i$ to the power of $1$ .

$- \frac{i}{i^{1} i}$

Step 5.3.2

Raise $i$ to the power of $1$ .

$- \frac{i}{i^{1} i^{1}}$

Step 5.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{i}{i^{1 + 1}}$

Step 5.3.4

Add $1$ and $1$ .

$- \frac{i}{i^{2}}$

Step 5.3.5

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

$- \frac{i}{- 1}$

Step 6

Move the negative one from the denominator of $\frac{i}{- 1}$ .

$- (- 1 \cdot i)$

Step 7

Rewrite $- 1 \cdot i$ as $- i$ .

$- - i$

Step 8

Multiply $- 1$ by $- 1$ .

$1 i$

Step 9

Multiply $i$ by $1$ .

$i$

Step 10

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 11

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 12

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

$| z | = \sqrt{1^{2}}$

Step 13

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

$| z | = 1$

Step 14

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

$θ = \arctan (\frac{1}{0})$

Step 15

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

$θ = \frac{π}{2}$

Step 16

Substitute the values of $θ = \frac{π}{2}$ and $| z | = 1$ .

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