Precalculus Examples

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

Step 1

Divide each term in $\tan (\frac{x}{2}) = \sin (\frac{x}{2})$ by $\tan (\frac{x}{2})$ and simplify.

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Step 1.1

Divide each term in $\tan (\frac{x}{2}) = \sin (\frac{x}{2})$ by $\tan (\frac{x}{2})$ .

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

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $\tan (\frac{x}{2})$ .

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Step 1.2.1.1

Cancel the common factor.

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

Step 1.2.1.2

Rewrite the expression.

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

Step 1.3

Simplify the right side.

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Step 1.3.1

Rewrite $\tan (\frac{x}{2})$ in terms of sines and cosines.

$1 = \frac{\sin (\frac{x}{2})}{\frac{\sin (\frac{x}{2})}{\cos (\frac{x}{2})}}$

Step 1.3.2

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (\frac{x}{2})}{\cos (\frac{x}{2})}$ .

$1 = \sin (\frac{x}{2}) \frac{\cos (\frac{x}{2})}{\sin (\frac{x}{2})}$

Step 1.3.3

Write $\sin (\frac{x}{2})$ as a fraction with denominator $1$ .

$1 = \frac{\sin (\frac{x}{2})}{1} \cdot \frac{\cos (\frac{x}{2})}{\sin (\frac{x}{2})}$

Step 1.3.4

Cancel the common factor of $\sin (\frac{x}{2})$ .

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Step 1.3.4.1

Cancel the common factor.

$1 = \frac{\sin (\frac{x}{2})}{1} \cdot \frac{\cos (\frac{x}{2})}{\sin (\frac{x}{2})}$

Step 1.3.4.2

Rewrite the expression.

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

Step 2

Rewrite the equation as $\cos (\frac{x}{2}) = 1$ .

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

Step 3

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

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

Step 4

Simplify the right side.

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Step 4.1

The exact value of $\arccos (1)$ is $0$ .

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

Step 5

Set the numerator equal to zero.

$x = 0$

Step 6

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$\frac{x}{2} = 2 π - 0$

Step 7

Solve for $x$ .

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Step 7.1

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (2 π - 0)$

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.

$2 \frac{x}{2} = 2 (2 π - 0)$

Step 7.2.1.1.2

Rewrite the expression.

$x = 2 (2 π - 0)$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Simplify $2 (2 π - 0)$ .

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Step 7.2.2.1.1

Subtract $0$ from $2 π$ .

$x = 2 (2 π)$

Step 7.2.2.1.2

Multiply $2$ by $2$ .

$x = 4 π$

Step 8

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

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Step 8.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 8.2

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

$\frac{2 π}{| \frac{1}{2} |}$

Step 8.3

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

$\frac{2 π}{\frac{1}{2}}$

Step 8.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 8.5

Multiply $2$ by $2$ .

$4 π$

Step 9

The period of the $\cos (\frac{x}{2})$ function is $4 π$ so values will repeat every $4 π$ radians in both directions.

$x = 4 π n, 4 π + 4 π n$ , for any integer $n$

Step 10

Consolidate the answers.

$x = 4 π n$ , for any integer $n$