Trigonometry Examples

$\tan (\frac{x}{2}) - 1 = 0$ , $0 < x < 2 π$

Step 1

Add $1$ to both sides of the equation.

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

Step 2

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

$\frac{x}{2} = \arctan (1)$

Step 3

Simplify the right side.

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Step 3.1

The exact value of $\arctan (1)$ is $\frac{π}{4}$ .

$\frac{x}{2} = \frac{π}{4}$

Step 4

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \frac{π}{4}$

Step 5

Simplify both sides of the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Cancel the common factor of $2$ .

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Step 5.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \frac{π}{4}$

Step 5.1.1.2

Rewrite the expression.

$x = 2 \frac{π}{4}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.1

Factor $2$ out of $4$ .

$x = 2 \frac{π}{2 (2)}$

Step 5.2.1.2

Cancel the common factor.

$x = 2 \frac{π}{2 \cdot 2}$

Step 5.2.1.3

Rewrite the expression.

$x = \frac{π}{2}$

Step 6

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.

$\frac{x}{2} = π + \frac{π}{4}$

Step 7

Solve for $x$ .

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Step 7.1

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (π + \frac{π}{4})$

Step 7.2

Simplify both sides of the equation.

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Step 7.2.1

Simplify the left side.

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Step 7.2.1.1

Cancel the common factor of $2$ .

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Step 7.2.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 (π + \frac{π}{4})$

Step 7.2.1.1.2

Rewrite the expression.

$x = 2 (π + \frac{π}{4})$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Simplify $2 (π + \frac{π}{4})$ .

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Step 7.2.2.1.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = 2 (π \cdot \frac{4}{4} + \frac{π}{4})$

Step 7.2.2.1.2

Simplify terms.

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Step 7.2.2.1.2.1

Combine $π$ and $\frac{4}{4}$ .

$x = 2 (\frac{π \cdot 4}{4} + \frac{π}{4})$

Step 7.2.2.1.2.2

Combine the numerators over the common denominator.

$x = 2 \frac{π \cdot 4 + π}{4}$

Step 7.2.2.1.2.3

Cancel the common factor of $2$ .

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Step 7.2.2.1.2.3.1

Factor $2$ out of $4$ .

$x = 2 \frac{π \cdot 4 + π}{2 (2)}$

Step 7.2.2.1.2.3.2

Cancel the common factor.

$x = 2 \frac{π \cdot 4 + π}{2 \cdot 2}$

Step 7.2.2.1.2.3.3

Rewrite the expression.

$x = \frac{π \cdot 4 + π}{2}$

Step 7.2.2.1.3

Simplify the numerator.

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Step 7.2.2.1.3.1

Move $4$ to the left of $π$ .

$x = \frac{4 \cdot π + π}{2}$

Step 7.2.2.1.3.2

Add $4 π$ and $π$ .

$x = \frac{5 π}{2}$

Step 8

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

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Step 8.1

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

$\frac{π}{| b |}$

Step 8.2

Replace $b$ with $\frac{1}{2}$ in the formula for period.

$\frac{π}{| \frac{1}{2} |}$

Step 8.3

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{π}{\frac{1}{2}}$

Step 8.4

Multiply the numerator by the reciprocal of the denominator.

$π \cdot 2$

Step 8.5

Move $2$ to the left of $π$ .

$2 π$

Step 9

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

$x = \frac{π}{2} + 2 π n, \frac{5 π}{2} + 2 π n$ , for any integer $n$

Step 10

Consolidate the answers.

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

Step 11

Plug in $0$ for $n$ and simplify to see if the solution is contained in $(0, 2 π)$ .

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Step 11.1

Plug in $0$ for $n$ .

$\frac{π}{2} + 2 π (0)$

Step 11.2

Simplify.

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Step 11.2.1

Multiply $2 π (0)$ .

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Step 11.2.1.1

Multiply $0$ by $2$ .

$\frac{π}{2} + 0 π$

Step 11.2.1.2

Multiply $0$ by $π$ .

$\frac{π}{2} + 0$

Step 11.2.2

Add $\frac{π}{2}$ and $0$ .

$\frac{π}{2}$

Step 11.3

The interval $(0, 2 π)$ contains $\frac{π}{2}$ .

$x = \frac{π}{2}$