Trigonometry Examples

$\sin (\frac{x}{2}) - \cos (\frac{x}{2}) = 0$

Step 1

Divide each term in the equation by $\cos (\frac{x}{2})$ .

$\frac{\sin (\frac{x}{2})}{\cos (\frac{x}{2})} + \frac{- \cos (\frac{x}{2})}{\cos (\frac{x}{2})} = \frac{0}{\cos (\frac{x}{2})}$

Step 2

Convert from $\frac{\sin (\frac{x}{2})}{\cos (\frac{x}{2})}$ to $\tan (\frac{x}{2})$ .

$\tan (\frac{x}{2}) + \frac{- \cos (\frac{x}{2})}{\cos (\frac{x}{2})} = \frac{0}{\cos (\frac{x}{2})}$

Step 3

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

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Step 3.1

Cancel the common factor.

$\tan (\frac{x}{2}) + \frac{- \cos (\frac{x}{2})}{\cos (\frac{x}{2})} = \frac{0}{\cos (\frac{x}{2})}$

Step 3.2

Divide $- 1$ by $1$ .

$\tan (\frac{x}{2}) - 1 = \frac{0}{\cos (\frac{x}{2})}$

Step 4

Separate fractions.

$\tan (\frac{x}{2}) - 1 = \frac{0}{1} \cdot \frac{1}{\cos (\frac{x}{2})}$

Step 5

Convert from $\frac{1}{\cos (\frac{x}{2})}$ to $\sec (\frac{x}{2})$ .

$\tan (\frac{x}{2}) - 1 = \frac{0}{1} \cdot \sec (\frac{x}{2})$

Step 6

Divide $0$ by $1$ .

$\tan (\frac{x}{2}) - 1 = 0 \sec (\frac{x}{2})$

Step 7

Multiply $0$ by $\sec (\frac{x}{2})$ .

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

Step 8

Add $1$ to both sides of the equation.

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

Step 9

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

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

Step 10

Simplify the right side.

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Step 10.1

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

$\frac{x}{2} = \frac{π}{4}$

Step 11

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \frac{π}{4}$

Step 12

Simplify both sides of the equation.

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Step 12.1

Simplify the left side.

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Step 12.1.1

Cancel the common factor of $2$ .

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Step 12.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \frac{π}{4}$

Step 12.1.1.2

Rewrite the expression.

$x = 2 \frac{π}{4}$

Step 12.2

Simplify the right side.

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Step 12.2.1

Cancel the common factor of $2$ .

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Step 12.2.1.1

Factor $2$ out of $4$ .

$x = 2 \frac{π}{2 (2)}$

Step 12.2.1.2

Cancel the common factor.

$x = 2 \frac{π}{2 \cdot 2}$

Step 12.2.1.3

Rewrite the expression.

$x = \frac{π}{2}$

Step 13

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 14

Solve for $x$ .

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Step 14.1

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (π + \frac{π}{4})$

Step 14.2

Simplify both sides of the equation.

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Step 14.2.1

Simplify the left side.

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Step 14.2.1.1

Cancel the common factor of $2$ .

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Step 14.2.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 (π + \frac{π}{4})$

Step 14.2.1.1.2

Rewrite the expression.

$x = 2 (π + \frac{π}{4})$

Step 14.2.2

Simplify the right side.

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Step 14.2.2.1

Simplify $2 (π + \frac{π}{4})$ .

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Step 14.2.2.1.1

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 14.2.2.1.2

Simplify terms.

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Step 14.2.2.1.2.1

Combine $π$ and $\frac{4}{4}$ .

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

Step 14.2.2.1.2.2

Combine the numerators over the common denominator.

$x = 2 \frac{π \cdot 4 + π}{4}$

Step 14.2.2.1.2.3

Cancel the common factor of $2$ .

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Step 14.2.2.1.2.3.1

Factor $2$ out of $4$ .

$x = 2 \frac{π \cdot 4 + π}{2 (2)}$

Step 14.2.2.1.2.3.2

Cancel the common factor.

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

Step 14.2.2.1.2.3.3

Rewrite the expression.

$x = \frac{π \cdot 4 + π}{2}$

Step 14.2.2.1.3

Simplify the numerator.

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Step 14.2.2.1.3.1

Move $4$ to the left of $π$ .

$x = \frac{4 \cdot π + π}{2}$

Step 14.2.2.1.3.2

Add $4 π$ and $π$ .

$x = \frac{5 π}{2}$

Step 15

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

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Step 15.1

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

$\frac{π}{| b |}$

Step 15.2

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

$\frac{π}{| \frac{1}{2} |}$

Step 15.3

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

$\frac{π}{\frac{1}{2}}$

Step 15.4

Multiply the numerator by the reciprocal of the denominator.

$π \cdot 2$

Step 15.5

Move $2$ to the left of $π$ .

$2 π$

Step 16

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 17

Consolidate the answers.

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