Trigonometry Examples

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

Step 1

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

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

Step 2

Multiply.

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Step 2.1

Combine.

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

Step 2.2

Multiply $1 - i$ by $1$ .

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

Step 2.3

Simplify the denominator.

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Step 2.3.1

Expand $(1 + 1 i) (1 - i)$ using the FOIL Method.

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Step 2.3.1.1

Apply the distributive property.

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

Step 2.3.1.2

Apply the distributive property.

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

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $1$ by $1$ .

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

Step 2.3.2.2

Multiply $- 1$ by $1$ .

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

Step 2.3.2.3

Multiply $1$ by $1$ .

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

Step 2.3.2.4

Multiply $- 1$ by $1$ .

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

Step 2.3.2.5

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

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

Step 2.3.2.6

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

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

Step 2.3.2.7

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

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

Step 2.3.2.8

Add $1$ and $1$ .

$- \frac{1 - i}{1 - 1 i + 1 i - i^{2}}$

Step 2.3.2.9

Add $- 1 i$ and $1 i$ .

$- \frac{1 - i}{1 + 0 - i^{2}}$

Step 2.3.2.10

Add $1$ and $0$ .

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

Step 2.3.3

Simplify each term.

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Step 2.3.3.1

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

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

Step 2.3.3.2

Multiply $- 1$ by $- 1$ .

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

Step 2.3.4

Add $1$ and $1$ .

$- \frac{1 - i}{2}$

Step 3

Split the fraction $\frac{1 - i}{2}$ into two fractions.

$- (\frac{1}{2} + \frac{- i}{2})$

Step 4

Move the negative in front of the fraction.

$- (\frac{1}{2} - \frac{i}{2})$

Step 5

Apply the distributive property.

$- \frac{1}{2} - - \frac{i}{2}$

Step 6

Multiply $- - \frac{i}{2}$ .

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Step 6.1

Multiply $- 1$ by $- 1$ .

$- \frac{1}{2} + 1 \frac{i}{2}$

Step 6.2

Multiply $\frac{i}{2}$ by $1$ .

$- \frac{1}{2} + \frac{i}{2}$

Step 7

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 8

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 9

Substitute the actual values of $a = - \frac{1}{2}$ and $b = \frac{1}{2}$ .

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

Step 10

Find $| z |$ .

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Step 10.1

Apply the product rule to $\frac{1}{2}$ .

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

Step 10.2

One to any power is one.

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

Step 10.3

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

$| z | = \sqrt{\frac{1}{4} + {(- \frac{1}{2})}^{2}}$

Step 10.4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 10.4.1

Apply the product rule to $- \frac{1}{2}$ .

$| z | = \sqrt{\frac{1}{4} + {(- 1)}^{2} {(\frac{1}{2})}^{2}}$

Step 10.4.2

Apply the product rule to $\frac{1}{2}$ .

$| z | = \sqrt{\frac{1}{4} + {(- 1)}^{2} (\frac{1^{2}}{2^{2}})}$

Step 10.5

Simplify the expression.

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Step 10.5.1

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

$| z | = \sqrt{\frac{1}{4} + 1 (\frac{1^{2}}{2^{2}})}$

Step 10.5.2

Multiply $\frac{1^{2}}{2^{2}}$ by $1$ .

$| z | = \sqrt{\frac{1}{4} + \frac{1^{2}}{2^{2}}}$

Step 10.5.3

One to any power is one.

$| z | = \sqrt{\frac{1}{4} + \frac{1}{2^{2}}}$

Step 10.5.4

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

$| z | = \sqrt{\frac{1}{4} + \frac{1}{4}}$

Step 10.5.5

Combine the numerators over the common denominator.

$| z | = \sqrt{\frac{1 + 1}{4}}$

Step 10.5.6

Add $1$ and $1$ .

$| z | = \sqrt{\frac{2}{4}}$

Step 10.6

Cancel the common factor of $2$ and $4$ .

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Step 10.6.1

Factor $2$ out of $2$ .

$| z | = \sqrt{\frac{2 (1)}{4}}$

Step 10.6.2

Cancel the common factors.

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Step 10.6.2.1

Factor $2$ out of $4$ .

$| z | = \sqrt{\frac{2 \cdot 1}{2 \cdot 2}}$

Step 10.6.2.2

Cancel the common factor.

$| z | = \sqrt{\frac{2 \cdot 1}{2 \cdot 2}}$

Step 10.6.2.3

Rewrite the expression.

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

Step 10.7

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

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

Step 10.8

Any root of $1$ is $1$ .

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

Step 10.9

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$| z | = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 10.10

Combine and simplify the denominator.

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Step 10.10.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 10.10.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 10.10.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 10.10.4

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

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

Step 10.10.5

Add $1$ and $1$ .

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

Step 10.10.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 10.10.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$| z | = \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 10.10.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 10.10.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 10.10.6.4

Cancel the common factor of $2$ .

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Step 10.10.6.4.1

Cancel the common factor.

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

Step 10.10.6.4.2

Rewrite the expression.

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

Step 10.10.6.5

Evaluate the exponent.

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

Step 11

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

$θ = \arctan (\frac{\frac{1}{2}}{- \frac{1}{2}})$

Step 12

Since inverse tangent of $\frac{\frac{1}{2}}{- \frac{1}{2}}$ produces an angle in the second quadrant, the value of the angle is $\frac{3 π}{4}$ .

$θ = \frac{3 π}{4}$

Step 13

Substitute the values of $θ = \frac{3 π}{4}$ and $| z | = \frac{\sqrt{2}}{2}$ .

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