Trigonometry Examples

$\sqrt{3} + i$

Step 1

Roots are the points where the graph intercepts with the x-axis $(y = 0)$ .

$y = 0$ at the roots

Step 2

The root at $x = \sqrt{3} + i$ was found by solving for $x$ when $x - (\sqrt{3} + i) = y$ and $y = 0$ .

The factor is $x - \sqrt{3} - i$

Step 3

The root at $x = \sqrt{3} - i$ was found by solving for $x$ when $x - (\sqrt{3} - i) = y$ and $y = 0$ .

The factor is $x - \sqrt{3} + i$

Step 4

Combine all the factors into a single equation.

$y = (x - \sqrt{3} - i) (x - \sqrt{3} + i)$

Step 5

Multiply all the factors to simplify the equation $y = (x - \sqrt{3} - i) (x - \sqrt{3} + i)$ .

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Step 5.1

Expand $(x - \sqrt{3} - i) (x - \sqrt{3} + i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x \cdot x + x (- \sqrt{3}) + x i - \sqrt{3} x - \sqrt{3} (- \sqrt{3}) - \sqrt{3} i - i x - i (- \sqrt{3}) - i i$

Step 5.2

Simplify terms.

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Step 5.2.1

Combine the opposite terms in $x \cdot x + x (- \sqrt{3}) + x i - \sqrt{3} x - \sqrt{3} (- \sqrt{3}) - \sqrt{3} i - i x - i (- \sqrt{3}) - i i$ .

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Step 5.2.1.1

Reorder the factors in the terms $x i$ and $- i x$ .

$y = x \cdot x + x (- \sqrt{3}) + i x - \sqrt{3} x - \sqrt{3} (- \sqrt{3}) - \sqrt{3} i - i x - i (- \sqrt{3}) - i i$

Step 5.2.1.2

Subtract $i x$ from $i x$ .

$y = x \cdot x + x (- \sqrt{3}) - \sqrt{3} x - \sqrt{3} (- \sqrt{3}) - \sqrt{3} i + 0 - i (- \sqrt{3}) - i i$

Step 5.2.1.3

Add $x \cdot x + x (- \sqrt{3}) - \sqrt{3} x - \sqrt{3} (- \sqrt{3}) - \sqrt{3} i$ and $0$ .

$y = x \cdot x + x (- \sqrt{3}) - \sqrt{3} x - \sqrt{3} (- \sqrt{3}) - \sqrt{3} i - i (- \sqrt{3}) - i i$

Step 5.2.2

Simplify each term.

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Step 5.2.2.1

Multiply $x$ by $x$ .

$y = x^{2} + x (- \sqrt{3}) - \sqrt{3} x - \sqrt{3} (- \sqrt{3}) - \sqrt{3} i - i (- \sqrt{3}) - i i$

Step 5.2.2.2

Multiply $- \sqrt{3} (- \sqrt{3})$ .

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Step 5.2.2.2.1

Multiply $- 1$ by $- 1$ .

$y = x^{2} + x (- \sqrt{3}) - \sqrt{3} x + 1 \sqrt{3} \sqrt{3} - \sqrt{3} i - i (- \sqrt{3}) - i i$

Step 5.2.2.2.2

Multiply $\sqrt{3}$ by $1$ .

$y = x^{2} + x (- \sqrt{3}) - \sqrt{3} x + \sqrt{3} \sqrt{3} - \sqrt{3} i - i (- \sqrt{3}) - i i$

Step 5.2.2.2.3

Raise $\sqrt{3}$ to the power of $1$ .

$y = x^{2} + x (- \sqrt{3}) - \sqrt{3} x + \sqrt{3} \sqrt{3} - \sqrt{3} i - i (- \sqrt{3}) - i i$

Step 5.2.2.2.4

Raise $\sqrt{3}$ to the power of $1$ .

$y = x^{2} + x (- \sqrt{3}) - \sqrt{3} x + \sqrt{3} \sqrt{3} - \sqrt{3} i - i (- \sqrt{3}) - i i$

Step 5.2.2.2.5

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

$y = x^{2} + x (- \sqrt{3}) - \sqrt{3} x + {\sqrt{3}}^{1 + 1} - \sqrt{3} i - i (- \sqrt{3}) - i i$

Step 5.2.2.2.6

Add $1$ and $1$ .

$y = x^{2} + x (- \sqrt{3}) - \sqrt{3} x + {\sqrt{3}}^{2} - \sqrt{3} i - i (- \sqrt{3}) - i i$

Step 5.2.2.3

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 5.2.2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$y = x^{2} + x (- \sqrt{3}) - \sqrt{3} x + {(3^{\frac{1}{2}})}^{2} - \sqrt{3} i - i (- \sqrt{3}) - i i$

Step 5.2.2.3.2

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

$y = x^{2} + x (- \sqrt{3}) - \sqrt{3} x + 3^{\frac{1}{2} \cdot 2} - \sqrt{3} i - i (- \sqrt{3}) - i i$

Step 5.2.2.3.3

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

$y = x^{2} + x (- \sqrt{3}) - \sqrt{3} x + 3^{\frac{2}{2}} - \sqrt{3} i - i (- \sqrt{3}) - i i$

Step 5.2.2.3.4

Cancel the common factor of $2$ .

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Step 5.2.2.3.4.1

Cancel the common factor.

$y = x^{2} + x (- \sqrt{3}) - \sqrt{3} x + 3^{\frac{2}{2}} - \sqrt{3} i - i (- \sqrt{3}) - i i$

Step 5.2.2.3.4.2

Rewrite the expression.

$y = x^{2} + x (- \sqrt{3}) - \sqrt{3} x + 3 - \sqrt{3} i - i (- \sqrt{3}) - i i$

Step 5.2.2.3.5

Evaluate the exponent.

$y = x^{2} + x (- \sqrt{3}) - \sqrt{3} x + 3 - \sqrt{3} i - i (- \sqrt{3}) - i i$

Step 5.2.2.4

Multiply $- i (- \sqrt{3})$ .

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Step 5.2.2.4.1

Multiply $- 1$ by $- 1$ .

$y = x^{2} + x (- \sqrt{3}) - \sqrt{3} x + 3 - \sqrt{3} i + 1 i \sqrt{3} - i i$

Step 5.2.2.4.2

Multiply $\sqrt{3}$ by $1$ .

$y = x^{2} + x (- \sqrt{3}) - \sqrt{3} x + 3 - \sqrt{3} i + \sqrt{3} i - i i$

Step 5.2.2.5

Multiply $- i i$ .

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Step 5.2.2.5.1

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

$y = x^{2} + x (- \sqrt{3}) - \sqrt{3} x + 3 - \sqrt{3} i + \sqrt{3} i - (i i)$

Step 5.2.2.5.2

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

$y = x^{2} + x (- \sqrt{3}) - \sqrt{3} x + 3 - \sqrt{3} i + \sqrt{3} i - (i i)$

Step 5.2.2.5.3

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

$y = x^{2} + x (- \sqrt{3}) - \sqrt{3} x + 3 - \sqrt{3} i + \sqrt{3} i - i^{1 + 1}$

Step 5.2.2.5.4

Add $1$ and $1$ .

$y = x^{2} + x (- \sqrt{3}) - \sqrt{3} x + 3 - \sqrt{3} i + \sqrt{3} i - i^{2}$

Step 5.2.2.6

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

$y = x^{2} + x (- \sqrt{3}) - \sqrt{3} x + 3 - \sqrt{3} i + \sqrt{3} i + 1$

Step 5.2.2.7

Multiply $- 1$ by $- 1$ .

$y = x^{2} + x (- \sqrt{3}) - \sqrt{3} x + 3 - \sqrt{3} i + \sqrt{3} i + 1$

Step 5.2.3

Simplify by adding terms.

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Step 5.2.3.1

Combine the opposite terms in $x^{2} + x (- \sqrt{3}) - \sqrt{3} x + 3 - \sqrt{3} i + \sqrt{3} i + 1$ .

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Step 5.2.3.1.1

Add $- \sqrt{3} i$ and $\sqrt{3} i$ .

$y = x^{2} + x (- \sqrt{3}) - \sqrt{3} x + 3 + 0 + 1$

Step 5.2.3.1.2

Add $x^{2} + x (- \sqrt{3}) - \sqrt{3} x + 3$ and $0$ .

$y = x^{2} + x (- \sqrt{3}) - \sqrt{3} x + 3 + 1$

Step 5.2.3.2

Reorder the factors of $- \sqrt{3} x$ .

$y = x^{2} + x (- \sqrt{3}) - x \sqrt{3} + 3 + 1$

Step 5.3

Subtract $x \sqrt{3}$ from $x (- \sqrt{3})$ .

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Step 5.3.1

Reorder $x$ and $- 1$ .

$y = x^{2} - 1 \cdot (x \sqrt{3}) - x \sqrt{3} + 3 + 1$

Step 5.3.2

Subtract $x \sqrt{3}$ from $- 1 \cdot x \sqrt{3}$ .

$y = x^{2} - 2 x \sqrt{3} + 3 + 1$

Step 5.4

Add $3$ and $1$ .

$y = x^{2} - 2 x \sqrt{3} + 4$

Step 6