Trigonometry Examples

$6 \csc^{2} (\frac{x}{2}) = 3$

Step 1

Divide each term in $6 \csc^{2} (\frac{x}{2}) = 3$ by $6$ and simplify.

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Step 1.1

Divide each term in $6 \csc^{2} (\frac{x}{2}) = 3$ by $6$ .

$\frac{6 \csc^{2} (\frac{x}{2})}{6} = \frac{3}{6}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $6$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{6 \csc^{2} (\frac{x}{2})}{6} = \frac{3}{6}$

Step 1.2.1.2

Divide $\csc^{2} (\frac{x}{2})$ by $1$ .

$\csc^{2} (\frac{x}{2}) = \frac{3}{6}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Cancel the common factor of $3$ and $6$ .

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Step 1.3.1.1

Factor $3$ out of $3$ .

$\csc^{2} (\frac{x}{2}) = \frac{3 (1)}{6}$

Step 1.3.1.2

Cancel the common factors.

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Step 1.3.1.2.1

Factor $3$ out of $6$ .

$\csc^{2} (\frac{x}{2}) = \frac{3 \cdot 1}{3 \cdot 2}$

Step 1.3.1.2.2

Cancel the common factor.

$\csc^{2} (\frac{x}{2}) = \frac{3 \cdot 1}{3 \cdot 2}$

Step 1.3.1.2.3

Rewrite the expression.

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

Step 2

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

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

Step 3

Simplify $\pm \sqrt{\frac{1}{2}}$ .

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Step 3.1

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

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

Step 3.2

Any root of $1$ is $1$ .

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

Step 3.3

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

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

Step 3.4

Combine and simplify the denominator.

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Step 3.4.1

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

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

Step 3.4.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 3.4.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 3.4.4

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

$\csc (\frac{x}{2}) = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 3.4.5

Add $1$ and $1$ .

$\csc (\frac{x}{2}) = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 3.4.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 3.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Step 3.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\csc (\frac{x}{2}) = \pm \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 3.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$\csc (\frac{x}{2}) = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 3.4.6.4

Cancel the common factor of $2$ .

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Step 3.4.6.4.1

Cancel the common factor.

$\csc (\frac{x}{2}) = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 3.4.6.4.2

Rewrite the expression.

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

Step 3.4.6.5

Evaluate the exponent.

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

$\csc (\frac{x}{2}) = - \frac{\sqrt{2}}{2}$

Step 4.3

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

$\csc (\frac{x}{2}) = \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}$

Step 5

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

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

$\csc (\frac{x}{2}) = - \frac{\sqrt{2}}{2}$

Step 6

Solve for $x$ in $\csc (\frac{x}{2}) = \frac{\sqrt{2}}{2}$ .

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Step 6.1

The range of cosecant is $y \leq - 1$ and $y \geq 1$ . Since $\frac{\sqrt{2}}{2}$ does not fall in this range, there is no solution.

No solution

Step 7

Solve for $x$ in $\csc (\frac{x}{2}) = - \frac{\sqrt{2}}{2}$ .

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Step 7.1

The range of cosecant is $y \leq - 1$ and $y \geq 1$ . Since $- \frac{\sqrt{2}}{2}$ does not fall in this range, there is no solution.

No solution