Precalculus Examples

$\tan (a) = \frac{400 + 50 \sqrt{2}}{50 \sqrt{2}}$

Step 1

Simplify $\frac{400 + 50 \sqrt{2}}{50 \sqrt{2}}$ .

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Step 1.1

Cancel the common factor of $400 + 50 \sqrt{2}$ and $50$ .

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Step 1.1.1

Factor $50$ out of $400$ .

$\tan (a) = \frac{50 \cdot 8 + 50 \sqrt{2}}{50 \sqrt{2}}$

Step 1.1.2

Factor $50$ out of $50 \sqrt{2}$ .

$\tan (a) = \frac{50 \cdot 8 + 50 (\sqrt{2})}{50 \sqrt{2}}$

Step 1.1.3

Factor $50$ out of $50 (8) + 50 (\sqrt{2})$ .

$\tan (a) = \frac{50 (8 + \sqrt{2})}{50 \sqrt{2}}$

Step 1.1.4

Cancel the common factors.

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Step 1.1.4.1

Factor $50$ out of $50 \sqrt{2}$ .

$\tan (a) = \frac{50 (8 + \sqrt{2})}{50 (\sqrt{2})}$

Step 1.1.4.2

Cancel the common factor.

$\tan (a) = \frac{50 (8 + \sqrt{2})}{50 \sqrt{2}}$

Step 1.1.4.3

Rewrite the expression.

$\tan (a) = \frac{8 + \sqrt{2}}{\sqrt{2}}$

Step 1.2

Multiply $\frac{8 + \sqrt{2}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\tan (a) = \frac{8 + \sqrt{2}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 1.3

Simplify terms.

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Step 1.3.1

Combine and simplify the denominator.

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Step 1.3.1.1

Multiply $\frac{8 + \sqrt{2}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\tan (a) = \frac{(8 + \sqrt{2}) \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 1.3.1.2

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

$\tan (a) = \frac{(8 + \sqrt{2}) \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 1.3.1.3

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

$\tan (a) = \frac{(8 + \sqrt{2}) \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 1.3.1.4

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

$\tan (a) = \frac{(8 + \sqrt{2}) \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 1.3.1.5

Add $1$ and $1$ .

$\tan (a) = \frac{(8 + \sqrt{2}) \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 1.3.1.6

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

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Step 1.3.1.6.1

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

$\tan (a) = \frac{(8 + \sqrt{2}) \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 1.3.1.6.2

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

$\tan (a) = \frac{(8 + \sqrt{2}) \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 1.3.1.6.3

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

$\tan (a) = \frac{(8 + \sqrt{2}) \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.3.1.6.4

Cancel the common factor of $2$ .

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Step 1.3.1.6.4.1

Cancel the common factor.

$\tan (a) = \frac{(8 + \sqrt{2}) \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.3.1.6.4.2

Rewrite the expression.

$\tan (a) = \frac{(8 + \sqrt{2}) \sqrt{2}}{2^{1}}$

Step 1.3.1.6.5

Evaluate the exponent.

$\tan (a) = \frac{(8 + \sqrt{2}) \sqrt{2}}{2}$

Step 1.3.2

Apply the distributive property.

$\tan (a) = \frac{8 \sqrt{2} + \sqrt{2} \sqrt{2}}{2}$

Step 1.3.3

Combine using the product rule for radicals.

$\tan (a) = \frac{8 \sqrt{2} + \sqrt{2 \cdot 2}}{2}$

Step 1.4

Simplify each term.

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Step 1.4.1

Multiply $2$ by $2$ .

$\tan (a) = \frac{8 \sqrt{2} + \sqrt{4}}{2}$

Step 1.4.2

Rewrite $4$ as $2^{2}$ .

$\tan (a) = \frac{8 \sqrt{2} + \sqrt{2^{2}}}{2}$

Step 1.4.3

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

$\tan (a) = \frac{8 \sqrt{2} + 2}{2}$

Step 1.5

Cancel the common factor of $8 \sqrt{2} + 2$ and $2$ .

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Step 1.5.1

Factor $2$ out of $8 \sqrt{2}$ .

$\tan (a) = \frac{2 (4 \sqrt{2}) + 2}{2}$

Step 1.5.2

Factor $2$ out of $2$ .

$\tan (a) = \frac{2 (4 \sqrt{2}) + 2 \cdot 1}{2}$

Step 1.5.3

Factor $2$ out of $2 (4 \sqrt{2}) + 2 (1)$ .

$\tan (a) = \frac{2 (4 \sqrt{2} + 1)}{2}$

Step 1.5.4

Cancel the common factors.

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Step 1.5.4.1

Factor $2$ out of $2$ .

$\tan (a) = \frac{2 (4 \sqrt{2} + 1)}{2 (1)}$

Step 1.5.4.2

Cancel the common factor.

$\tan (a) = \frac{2 (4 \sqrt{2} + 1)}{2 \cdot 1}$

Step 1.5.4.3

Rewrite the expression.

$\tan (a) = \frac{4 \sqrt{2} + 1}{1}$

Step 1.5.4.4

Divide $4 \sqrt{2} + 1$ by $1$ .

$\tan (a) = 4 \sqrt{2} + 1$

Step 2

Convert the right side of the equation to its decimal equivalent.

$\tan (a) = 6.65685424$

Step 3

Take the inverse tangent of both sides of the equation to extract $a$ from inside the tangent.

$a = \arctan (6.65685424)$

Step 4

Simplify the right side.

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Step 4.1

Evaluate $\arctan (6.65685424)$ .

$a = 1.42169014$

Step 5

The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$a = (3.14159265) + 1.42169014$

Step 6

Solve for $a$ .

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Step 6.1

Remove parentheses.

$a = 3.14159265 + 1.42169014$

Step 6.2

Remove parentheses.

$a = (3.14159265) + 1.42169014$

Step 6.3

Add $3.14159265$ and $1.42169014$ .

$a = 4.5632828$

Step 7

Find the period of $\tan (a)$ .

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Step 7.1

The period of the function can be calculated using $\frac{π}{| b |}$ .

$\frac{π}{| b |}$

Step 7.2

Replace $b$ with $1$ in the formula for period.

$\frac{π}{| 1 |}$

Step 7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Step 7.4

Divide $π$ by $1$ .

$π$

Step 8

The period of the $\tan (a)$ function is $π$ so values will repeat every $π$ radians in both directions.

$a = 1.42169014 + π n, 4.5632828 + π n$ , for any integer $n$

Step 9

Consolidate $1.42169014 + π n$ and $4.5632828 + π n$ to $1.42169014 + π n$ .

$a = 1.42169014 + π n$ , for any integer $n$