Trigonometry Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Simplify the left side.

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Step 1.2.1

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

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Step 1.2.1.1

Cancel the common factor.

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

Step 1.2.1.2

Rewrite the expression.

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

Step 1.3

Simplify the right side.

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Step 1.3.1

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

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

Step 1.3.2

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (\frac{x}{2})}$ .

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

Step 1.3.3

Simplify.

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Step 1.3.3.1

Raise $\sin (\frac{x}{2})$ to the power of $1$ .

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

Step 1.3.3.2

Raise $\sin (\frac{x}{2})$ to the power of $1$ .

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

Step 1.3.3.3

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

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

Step 1.3.3.4

Add $1$ and $1$ .

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

Step 2

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

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

Step 3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

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

Step 4

Any root of $1$ is $1$ .

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

Step 5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 5.2

Next, use the negative value of the $\pm$ to find the second solution.

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

Step 5.3

The complete solution is the result of both the positive and negative portions of the solution.

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

Step 6

Set up each of the solutions to solve for $x$ .

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

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

Step 7

Solve for $x$ in $\sin (\frac{x}{2}) = 1$ .

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Step 7.1

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

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

Step 7.2

Simplify the right side.

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Step 7.2.1

The exact value of $\arcsin (1)$ is $\frac{π}{2}$ .

$\frac{x}{2} = \frac{π}{2}$

Step 7.3

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$x = π$

Step 7.4

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

$\frac{x}{2} = π - \frac{π}{2}$

Step 7.5

Solve for $x$ .

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Step 7.5.1

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (π - \frac{π}{2})$

Step 7.5.2

Simplify both sides of the equation.

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Step 7.5.2.1

Simplify the left side.

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Step 7.5.2.1.1

Cancel the common factor of $2$ .

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Step 7.5.2.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 (π - \frac{π}{2})$

Step 7.5.2.1.1.2

Rewrite the expression.

$x = 2 (π - \frac{π}{2})$

Step 7.5.2.2

Simplify the right side.

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Step 7.5.2.2.1

Simplify $2 (π - \frac{π}{2})$ .

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Step 7.5.2.2.1.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 7.5.2.2.1.2

Simplify terms.

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Step 7.5.2.2.1.2.1

Combine $π$ and $\frac{2}{2}$ .

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

Step 7.5.2.2.1.2.2

Combine the numerators over the common denominator.

$x = 2 \frac{π \cdot 2 - π}{2}$

Step 7.5.2.2.1.2.3

Cancel the common factor of $2$ .

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Step 7.5.2.2.1.2.3.1

Cancel the common factor.

$x = 2 \frac{π \cdot 2 - π}{2}$

Step 7.5.2.2.1.2.3.2

Rewrite the expression.

$x = π \cdot 2 - π$

Step 7.5.2.2.1.3

Move $2$ to the left of $π$ .

$x = 2 π - π$

Step 7.5.2.2.1.4

Subtract $π$ from $2 π$ .

$x = π$

Step 7.6

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

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Step 7.6.1

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

$\frac{2 π}{| b |}$

Step 7.6.2

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

$\frac{2 π}{| \frac{1}{2} |}$

Step 7.6.3

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

$\frac{2 π}{\frac{1}{2}}$

Step 7.6.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 7.6.5

Multiply $2$ by $2$ .

$4 π$

Step 7.7

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

$x = π + 4 π n$ , for any integer $n$

Step 8

Solve for $x$ in $\sin (\frac{x}{2}) = - 1$ .

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Step 8.1

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

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

Step 8.2

Simplify the right side.

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Step 8.2.1

The exact value of $\arcsin (- 1)$ is $- \frac{π}{2}$ .

$\frac{x}{2} = - \frac{π}{2}$

Step 8.3

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$x = - π$

Step 8.4

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$\frac{x}{2} = 2 π + \frac{π}{2} + π$

Step 8.5

Simplify the expression to find the second solution.

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Step 8.5.1

Subtract $2 π$ from $2 π + \frac{π}{2} + π$ .

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

Step 8.5.2

The resulting angle of $\frac{3 π}{2}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{2} + π$ .

$\frac{x}{2} = \frac{3 π}{2}$

Step 8.5.3

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$x = 3 π$

Step 8.6

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

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Step 8.6.1

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

$\frac{2 π}{| b |}$

Step 8.6.2

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

$\frac{2 π}{| \frac{1}{2} |}$

Step 8.6.3

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

$\frac{2 π}{\frac{1}{2}}$

Step 8.6.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 8.6.5

Multiply $2$ by $2$ .

$4 π$

Step 8.7

Add $4 π$ to every negative angle to get positive angles.

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Step 8.7.1

Add $4 π$ to $- π$ to find the positive angle.

$- π + 4 π$

Step 8.7.2

Subtract $π$ from $4 π$ .

$3 π$

Step 8.7.3

List the new angles.

$x = 3 π$

Step 8.8

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

$x = 3 π + 4 π n, 3 π + 4 π n$ , for any integer $n$

Step 9

List all of the solutions.

$x = π + 4 π n, 3 π + 4 π n, 3 π + 4 π n$ , for any integer $n$

Step 10

Consolidate the answers.

$x = π + 2 π n$ , for any integer $n$