Trigonometry Examples

$f (x) = {\tan (2 x)}^{\frac{1}{2}}$

Step 1

Convert expressions with fractional exponents to radicals.

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Step 1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\sqrt{\tan^{1} (2 x)}$

Step 1.2

Anything raised to $1$ is the base itself.

$\sqrt{\tan (2 x)}$

Step 2

Set the radicand in $\sqrt{\tan (2 x)}$ greater than or equal to $0$ to find where the expression is defined.

$\tan (2 x) \geq 0$

Step 3

Solve for $x$ .

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Step 3.1

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

$2 x \geq \arctan (0)$

Step 3.2

Simplify the right side.

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Step 3.2.1

The exact value of $\arctan (0)$ is $0$ .

$2 x \geq 0$

Step 3.3

Divide each term in $2 x \geq 0$ by $2$ and simplify.

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Step 3.3.1

Divide each term in $2 x \geq 0$ by $2$ .

$\frac{2 x}{2} \geq \frac{0}{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{0}{2}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{0}{2}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $0$ by $2$ .

$x \geq 0$

Step 3.4

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.

$2 x = π + 0$

Step 3.5

Solve for $x$ .

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Step 3.5.1

Add $π$ and $0$ .

$2 x = π$

Step 3.5.2

Divide each term in $2 x = π$ by $2$ and simplify.

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Step 3.5.2.1

Divide each term in $2 x = π$ by $2$ .

$\frac{2 x}{2} = \frac{π}{2}$

Step 3.5.2.2

Simplify the left side.

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Step 3.5.2.2.1

Cancel the common factor of $2$ .

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Step 3.5.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{π}{2}$

Step 3.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{2}$

Step 3.6

Find the period of $\tan (2 x)$ .

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Step 3.6.1

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

$\frac{π}{| b |}$

Step 3.6.2

Replace $b$ with $2$ in the formula for period.

$\frac{π}{| 2 |}$

Step 3.6.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{π}{2}$

Step 3.7

The period of the $\tan (2 x)$ function is $\frac{π}{2}$ so values will repeat every $\frac{π}{2}$ radians in both directions.

$x = \frac{π n}{2}, \frac{π}{2} + \frac{π n}{2}$ , for any integer $n$

Step 3.8

Consolidate the answers.

$x = \frac{π n}{2}$ , for any integer $n$

Step 3.9

Find the domain of $\tan (2 x)$ .

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Step 3.9.1

Set the argument in $\tan (2 x)$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$2 x = \frac{π}{2} + π n$ , for any integer $n$

Step 3.9.2

Divide each term in $2 x = \frac{π}{2} + π n$ by $2$ and simplify.

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Step 3.9.2.1

Divide each term in $2 x = \frac{π}{2} + π n$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{π}{2}}{2} + \frac{π n}{2}$

Step 3.9.2.2

Simplify the left side.

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Step 3.9.2.2.1

Cancel the common factor of $2$ .

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Step 3.9.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{π}{2}}{2} + \frac{π n}{2}$

Step 3.9.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{2} + \frac{π n}{2}$

Step 3.9.2.3

Simplify the right side.

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Step 3.9.2.3.1

Simplify each term.

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Step 3.9.2.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{2} + \frac{π n}{2}$

Step 3.9.2.3.1.2

Multiply $\frac{π}{2} \cdot \frac{1}{2}$ .

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Step 3.9.2.3.1.2.1

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

$x = \frac{π}{2 \cdot 2} + \frac{π n}{2}$

Step 3.9.2.3.1.2.2

Multiply $2$ by $2$ .

$x = \frac{π}{4} + \frac{π n}{2}$

Step 3.9.3

The domain is all values of $x$ that make the expression defined.

${x | x \neq \frac{π}{4} + \frac{π n}{2}}$ $n$ , for any integer $n$

Step 3.10

Use each root to create test intervals.

$0 < x < \frac{π}{4}$

$\frac{π}{4} < x < \frac{π}{2}$

Step 3.11

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 3.11.1

Test a value on the interval $0 < x < \frac{π}{4}$ to see if it makes the inequality true.

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Step 3.11.1.1

Choose a value on the interval $0 < x < \frac{π}{4}$ and see if this value makes the original inequality true.

$x = 0.39$

Step 3.11.1.2

Replace $x$ with $0.39$ in the original inequality.

$\tan (2 (0.39)) \geq 0$

Step 3.11.1.3

The left side $0.98926153$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 3.11.2

Test a value on the interval $\frac{π}{4} < x < \frac{π}{2}$ to see if it makes the inequality true.

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Step 3.11.2.1

Choose a value on the interval $\frac{π}{4} < x < \frac{π}{2}$ and see if this value makes the original inequality true.

$x = 1$

Step 3.11.2.2

Replace $x$ with $1$ in the original inequality.

$\tan (2 (1)) \geq 0$

Step 3.11.2.3

The left side $- 2.18503986$ is less than the right side $0$ , which means that the given statement is false.

False

Step 3.11.3

Compare the intervals to determine which ones satisfy the original inequality.

$0 < x < \frac{π}{4}$ True

$\frac{π}{4} < x < \frac{π}{2}$ False

$0 < x < \frac{π}{4}$ True

$\frac{π}{4} < x < \frac{π}{2}$ False

Step 3.12

The solution consists of all of the true intervals.

$0 + \frac{π n}{2} \leq x < \frac{π}{4} + \frac{π n}{2}$ , for any integer $n$

Step 4

Set the argument in $\tan (2 x)$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$2 x = \frac{π}{2} + π n$ , for any integer $n$

Step 5

Divide each term in $2 x = \frac{π}{2} + π n$ by $2$ and simplify.

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Step 5.1

Divide each term in $2 x = \frac{π}{2} + π n$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{π}{2}}{2} + \frac{π n}{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{π}{2}}{2} + \frac{π n}{2}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{2} + \frac{π n}{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{2} + \frac{π n}{2}$

Step 5.3.1.2

Multiply $\frac{π}{2} \cdot \frac{1}{2}$ .

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Step 5.3.1.2.1

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

$x = \frac{π}{2 \cdot 2} + \frac{π n}{2}$

Step 5.3.1.2.2

Multiply $2$ by $2$ .

$x = \frac{π}{4} + \frac{π n}{2}$

Step 6

The domain is all values of $x$ that make the expression defined.

Set-Builder Notation:

${x | x = 0 + \frac{π n}{2}, x = \frac{π}{4} + \frac{π n}{2}, x \neq \frac{π}{4} + \frac{π n}{2}}$

Step 7

Determine the domain and range.

Domain: ${x | x = 0 + \frac{π n}{2}, x = \frac{π}{4} + \frac{π n}{2}, x \neq \frac{π}{4} + \frac{π n}{2}}$

Range:

Step 8