Trigonometry Examples

$\sec (\frac{5 π}{3} - \frac{3 π}{4}) = \csc (x)$

Step 1

Rewrite the equation as $\csc (x) = \sec (\frac{5 π}{3} - \frac{3 π}{4})$ .

$\csc (x) = \sec (\frac{5 π}{3} - \frac{3 π}{4})$

Step 2

Simplify $\sec (\frac{5 π}{3} - \frac{3 π}{4})$ .

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Step 2.1

To write $\frac{5 π}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\csc (x) = \sec (\frac{5 π}{3} \cdot \frac{4}{4} - \frac{3 π}{4})$

Step 2.2

To write $- \frac{3 π}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\csc (x) = \sec (\frac{5 π}{3} \cdot \frac{4}{4} - \frac{3 π}{4} \cdot \frac{3}{3})$

Step 2.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{5 π}{3}$ by $\frac{4}{4}$ .

$\csc (x) = \sec (\frac{5 π \cdot 4}{3 \cdot 4} - \frac{3 π}{4} \cdot \frac{3}{3})$

Step 2.3.2

Multiply $3$ by $4$ .

$\csc (x) = \sec (\frac{5 π \cdot 4}{12} - \frac{3 π}{4} \cdot \frac{3}{3})$

Step 2.3.3

Multiply $\frac{3 π}{4}$ by $\frac{3}{3}$ .

$\csc (x) = \sec (\frac{5 π \cdot 4}{12} - \frac{3 π \cdot 3}{4 \cdot 3})$

Step 2.3.4

Multiply $4$ by $3$ .

$\csc (x) = \sec (\frac{5 π \cdot 4}{12} - \frac{3 π \cdot 3}{12})$

Step 2.4

Combine the numerators over the common denominator.

$\csc (x) = \sec (\frac{5 π \cdot 4 - 3 π \cdot 3}{12})$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $4$ by $5$ .

$\csc (x) = \sec (\frac{20 π - 3 π \cdot 3}{12})$

Step 2.5.2

Multiply $3$ by $- 3$ .

$\csc (x) = \sec (\frac{20 π - 9 π}{12})$

Step 2.5.3

Subtract $9 π$ from $20 π$ .

$\csc (x) = \sec (\frac{11 π}{12})$

Step 2.6

The exact value of $\sec (\frac{11 π}{12})$ is $- \sqrt{6} + \sqrt{2}$ .

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Step 2.6.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because secant is negative in the second quadrant.

$\csc (x) = - \sec (\frac{π}{12})$

Step 2.6.2

Split $\frac{π}{12}$ into two angles where the values of the six trigonometric functions are known.

$\csc (x) = - \sec (\frac{π}{4} - \frac{π}{6})$

Step 2.6.3

Apply the difference of angles identity.

$\csc (x) = - \frac{\sec (\frac{π}{4}) \sec (\frac{π}{6}) \csc (\frac{π}{4}) \csc (\frac{π}{6})}{\csc (\frac{π}{4}) \csc (\frac{π}{6}) + \sec (\frac{π}{4}) \sec (\frac{π}{6})}$

Step 2.6.4

The exact value of $\sec (\frac{π}{4})$ is $\frac{2}{\sqrt{2}}$ .

$\csc (x) = - \frac{\frac{2}{\sqrt{2}} \sec (\frac{π}{6}) \csc (\frac{π}{4}) \csc (\frac{π}{6})}{\csc (\frac{π}{4}) \csc (\frac{π}{6}) + \sec (\frac{π}{4}) \sec (\frac{π}{6})}$

Step 2.6.5

The exact value of $\sec (\frac{π}{6})$ is $\frac{2}{\sqrt{3}}$ .

$\csc (x) = - \frac{\frac{2}{\sqrt{2}} \cdot \frac{2}{\sqrt{3}} \csc (\frac{π}{4}) \csc (\frac{π}{6})}{\csc (\frac{π}{4}) \csc (\frac{π}{6}) + \sec (\frac{π}{4}) \sec (\frac{π}{6})}$

Step 2.6.6

The exact value of $\csc (\frac{π}{4})$ is $\sqrt{2}$ .

$\csc (x) = - \frac{\frac{2}{\sqrt{2}} \cdot \frac{2}{\sqrt{3}} \sqrt{2} \csc (\frac{π}{6})}{\csc (\frac{π}{4}) \csc (\frac{π}{6}) + \sec (\frac{π}{4}) \sec (\frac{π}{6})}$

Step 2.6.7

The exact value of $\csc (\frac{π}{6})$ is $2$ .

$\csc (x) = - \frac{\frac{2}{\sqrt{2}} \cdot \frac{2}{\sqrt{3}} \sqrt{2} \cdot 2}{\csc (\frac{π}{4}) \csc (\frac{π}{6}) + \sec (\frac{π}{4}) \sec (\frac{π}{6})}$

Step 2.6.8

The exact value of $\csc (\frac{π}{4})$ is $\sqrt{2}$ .

$\csc (x) = - \frac{\frac{2}{\sqrt{2}} \cdot \frac{2}{\sqrt{3}} \sqrt{2} \cdot 2}{\sqrt{2} \csc (\frac{π}{6}) + \sec (\frac{π}{4}) \sec (\frac{π}{6})}$

Step 2.6.9

The exact value of $\csc (\frac{π}{6})$ is $2$ .

$\csc (x) = - \frac{\frac{2}{\sqrt{2}} \cdot \frac{2}{\sqrt{3}} \sqrt{2} \cdot 2}{\sqrt{2} \cdot 2 + \sec (\frac{π}{4}) \sec (\frac{π}{6})}$

Step 2.6.10

The exact value of $\sec (\frac{π}{4})$ is $\frac{2}{\sqrt{2}}$ .

$\csc (x) = - \frac{\frac{2}{\sqrt{2}} \cdot \frac{2}{\sqrt{3}} \sqrt{2} \cdot 2}{\sqrt{2} \cdot 2 + \frac{2}{\sqrt{2}} \sec (\frac{π}{6})}$

Step 2.6.11

The exact value of $\sec (\frac{π}{6})$ is $\frac{2}{\sqrt{3}}$ .

$\csc (x) = - \frac{\frac{2}{\sqrt{2}} \cdot \frac{2}{\sqrt{3}} \sqrt{2} \cdot 2}{\sqrt{2} \cdot 2 + \frac{2}{\sqrt{2}} \cdot \frac{2}{\sqrt{3}}}$

Step 2.6.12

Simplify $- \frac{\frac{2}{\sqrt{2}} \cdot \frac{2}{\sqrt{3}} \sqrt{2} \cdot 2}{\sqrt{2} \cdot 2 + \frac{2}{\sqrt{2}} \cdot \frac{2}{\sqrt{3}}}$ .

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Step 2.6.12.1

Simplify the numerator.

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Step 2.6.12.1.1

Multiply $\frac{2}{\sqrt{2}}$ by $\frac{2}{\sqrt{3}}$ .

$\csc (x) = - \frac{\frac{2 \cdot 2}{\sqrt{2} \sqrt{3}} \sqrt{2} \cdot 2}{\sqrt{2} \cdot 2 + \frac{2}{\sqrt{2}} \cdot \frac{2}{\sqrt{3}}}$

Step 2.6.12.1.2

Combine $\frac{2 \cdot 2}{\sqrt{2} \sqrt{3}}$ and $\sqrt{2}$ .

$\csc (x) = - \frac{\frac{2 \cdot 2 \sqrt{2}}{\sqrt{2} \sqrt{3}} \cdot 2}{\sqrt{2} \cdot 2 + \frac{2}{\sqrt{2}} \cdot \frac{2}{\sqrt{3}}}$

Step 2.6.12.1.3

Combine $\frac{2 \cdot 2 \sqrt{2}}{\sqrt{2} \sqrt{3}}$ and $2$ .

$\csc (x) = - \frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{\sqrt{2} \cdot 2 + \frac{2}{\sqrt{2}} \cdot \frac{2}{\sqrt{3}}}$

Step 2.6.12.2

Simplify the denominator.

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Step 2.6.12.2.1

Move $2$ to the left of $\sqrt{2}$ .

$\csc (x) = - \frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{2 \cdot \sqrt{2} + \frac{2}{\sqrt{2}} \cdot \frac{2}{\sqrt{3}}}$

Step 2.6.12.2.2

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

$\csc (x) = - \frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{2 \sqrt{2} + \frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} \frac{2}{\sqrt{3}}}$

Step 2.6.12.2.3

Combine and simplify the denominator.

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Step 2.6.12.2.3.1

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

$\csc (x) = - \frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{2 \sqrt{2} + \frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}} \cdot \frac{2}{\sqrt{3}}}$

Step 2.6.12.2.3.2

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

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

Step 2.6.12.2.3.3

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

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

Step 2.6.12.2.3.4

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

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

Step 2.6.12.2.3.5

Add $1$ and $1$ .

$\csc (x) = - \frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{2 \sqrt{2} + \frac{2 \sqrt{2}}{{\sqrt{2}}^{2}} \cdot \frac{2}{\sqrt{3}}}$

Step 2.6.12.2.3.6

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

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Step 2.6.12.2.3.6.1

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

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

Step 2.6.12.2.3.6.2

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

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

Step 2.6.12.2.3.6.3

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

$\csc (x) = - \frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{2 \sqrt{2} + \frac{2 \sqrt{2}}{2^{\frac{2}{2}}} \cdot \frac{2}{\sqrt{3}}}$

Step 2.6.12.2.3.6.4

Cancel the common factor of $2$ .

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Step 2.6.12.2.3.6.4.1

Cancel the common factor.

$\csc (x) = - \frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{2 \sqrt{2} + \frac{2 \sqrt{2}}{2^{\frac{2}{2}}} \cdot \frac{2}{\sqrt{3}}}$

Step 2.6.12.2.3.6.4.2

Rewrite the expression.

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

Step 2.6.12.2.3.6.5

Evaluate the exponent.

$\csc (x) = - \frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{2 \sqrt{2} + \frac{2 \sqrt{2}}{2} \cdot \frac{2}{\sqrt{3}}}$

Step 2.6.12.2.4

Cancel the common factor of $2$ .

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Step 2.6.12.2.4.1

Cancel the common factor.

$\csc (x) = - \frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{2 \sqrt{2} + \frac{2 \sqrt{2}}{2} \cdot \frac{2}{\sqrt{3}}}$

Step 2.6.12.2.4.2

Rewrite the expression.

$\csc (x) = - \frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{2 \sqrt{2} + 2 \sqrt{2} \frac{1}{\sqrt{3}}}$

Step 2.6.12.2.5

Combine $\frac{1}{\sqrt{3}}$ and $2$ .

$\csc (x) = - \frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{2 \sqrt{2} + \frac{2}{\sqrt{3}} \sqrt{2}}$

Step 2.6.12.2.6

Combine $\frac{2}{\sqrt{3}}$ and $\sqrt{2}$ .

$\csc (x) = - \frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{2 \sqrt{2} + \frac{2 \sqrt{2}}{\sqrt{3}}}$

Step 2.6.12.2.7

Multiply $\frac{2 \sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\csc (x) = - \frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{2 \sqrt{2} + \frac{2 \sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}}$

Step 2.6.12.2.8

Combine and simplify the denominator.

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Step 2.6.12.2.8.1

Multiply $\frac{2 \sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\csc (x) = - \frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{2 \sqrt{2} + \frac{2 \sqrt{2} \sqrt{3}}{\sqrt{3} \sqrt{3}}}$

Step 2.6.12.2.8.2

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 2.6.12.2.8.3

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 2.6.12.2.8.4

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

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

Step 2.6.12.2.8.5

Add $1$ and $1$ .

$\csc (x) = - \frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{2 \sqrt{2} + \frac{2 \sqrt{2} \sqrt{3}}{{\sqrt{3}}^{2}}}$

Step 2.6.12.2.8.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.6.12.2.8.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 2.6.12.2.8.6.2

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

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

Step 2.6.12.2.8.6.3

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

$\csc (x) = - \frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{2 \sqrt{2} + \frac{2 \sqrt{2} \sqrt{3}}{3^{\frac{2}{2}}}}$

Step 2.6.12.2.8.6.4

Cancel the common factor of $2$ .

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Step 2.6.12.2.8.6.4.1

Cancel the common factor.

$\csc (x) = - \frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{2 \sqrt{2} + \frac{2 \sqrt{2} \sqrt{3}}{3^{\frac{2}{2}}}}$

Step 2.6.12.2.8.6.4.2

Rewrite the expression.

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

Step 2.6.12.2.8.6.5

Evaluate the exponent.

$\csc (x) = - \frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{2 \sqrt{2} + \frac{2 \sqrt{2} \sqrt{3}}{3}}$

Step 2.6.12.2.9

Simplify the numerator.

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Step 2.6.12.2.9.1

Combine using the product rule for radicals.

$\csc (x) = - \frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{2 \sqrt{2} + \frac{2 \sqrt{3 \cdot 2}}{3}}$

Step 2.6.12.2.9.2

Multiply $3$ by $2$ .

$\csc (x) = - \frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{2 \sqrt{2} + \frac{2 \sqrt{6}}{3}}$

Step 2.6.12.2.10

To write $2 \sqrt{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\csc (x) = - \frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{2 \sqrt{2} \cdot \frac{3}{3} + \frac{2 \sqrt{6}}{3}}$

Step 2.6.12.2.11

Combine $2 \sqrt{2}$ and $\frac{3}{3}$ .

$\csc (x) = - \frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{\frac{2 \sqrt{2} \cdot 3}{3} + \frac{2 \sqrt{6}}{3}}$

Step 2.6.12.2.12

Combine the numerators over the common denominator.

$\csc (x) = - \frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{\frac{2 \sqrt{2} \cdot 3 + 2 \sqrt{6}}{3}}$

Step 2.6.12.2.13

Multiply $3$ by $2$ .

$\csc (x) = - \frac{\frac{2 \cdot 2 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{\frac{6 \sqrt{2} + 2 \sqrt{6}}{3}}$

Step 2.6.12.3

Simplify the numerator.

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Step 2.6.12.3.1

Multiply $2$ by $2$ .

$\csc (x) = - \frac{\frac{4 \sqrt{2} \cdot 2}{\sqrt{2} \sqrt{3}}}{\frac{6 \sqrt{2} + 2 \sqrt{6}}{3}}$

Step 2.6.12.3.2

Multiply $2$ by $4$ .

$\csc (x) = - \frac{\frac{8 \sqrt{2}}{\sqrt{2} \sqrt{3}}}{\frac{6 \sqrt{2} + 2 \sqrt{6}}{3}}$

Step 2.6.12.4

Simplify the denominator.

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Step 2.6.12.4.1

Combine using the product rule for radicals.

$\csc (x) = - \frac{\frac{8 \sqrt{2}}{\sqrt{2 \cdot 3}}}{\frac{6 \sqrt{2} + 2 \sqrt{6}}{3}}$

Step 2.6.12.4.2

Multiply $2$ by $3$ .

$\csc (x) = - \frac{\frac{8 \sqrt{2}}{\sqrt{6}}}{\frac{6 \sqrt{2} + 2 \sqrt{6}}{3}}$

Step 2.6.12.5

Simplify the numerator.

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Step 2.6.12.5.1

Combine $\sqrt{2}$ and $\sqrt{6}$ into a single radical.

$\csc (x) = - \frac{8 \sqrt{\frac{2}{6}}}{\frac{6 \sqrt{2} + 2 \sqrt{6}}{3}}$

Step 2.6.12.5.2

Cancel the common factor of $2$ and $6$ .

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Step 2.6.12.5.2.1

Factor $2$ out of $2$ .

$\csc (x) = - \frac{8 \sqrt{\frac{2 (1)}{6}}}{\frac{6 \sqrt{2} + 2 \sqrt{6}}{3}}$

Step 2.6.12.5.2.2

Cancel the common factors.

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Step 2.6.12.5.2.2.1

Factor $2$ out of $6$ .

$\csc (x) = - \frac{8 \sqrt{\frac{2 \cdot 1}{2 \cdot 3}}}{\frac{6 \sqrt{2} + 2 \sqrt{6}}{3}}$

Step 2.6.12.5.2.2.2

Cancel the common factor.

$\csc (x) = - \frac{8 \sqrt{\frac{2 \cdot 1}{2 \cdot 3}}}{\frac{6 \sqrt{2} + 2 \sqrt{6}}{3}}$

Step 2.6.12.5.2.2.3

Rewrite the expression.

$\csc (x) = - \frac{8 \sqrt{\frac{1}{3}}}{\frac{6 \sqrt{2} + 2 \sqrt{6}}{3}}$

Step 2.6.12.5.3

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

$\csc (x) = - \frac{8 \frac{\sqrt{1}}{\sqrt{3}}}{\frac{6 \sqrt{2} + 2 \sqrt{6}}{3}}$

Step 2.6.12.5.4

Any root of $1$ is $1$ .

$\csc (x) = - \frac{8 \frac{1}{\sqrt{3}}}{\frac{6 \sqrt{2} + 2 \sqrt{6}}{3}}$

Step 2.6.12.5.5

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\csc (x) = - \frac{8 (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})}{\frac{6 \sqrt{2} + 2 \sqrt{6}}{3}}$

Step 2.6.12.5.6

Combine and simplify the denominator.

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Step 2.6.12.5.6.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\csc (x) = - \frac{8 \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}}{\frac{6 \sqrt{2} + 2 \sqrt{6}}{3}}$

Step 2.6.12.5.6.2

Raise $\sqrt{3}$ to the power of $1$ .

$\csc (x) = - \frac{8 \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}}{\frac{6 \sqrt{2} + 2 \sqrt{6}}{3}}$

Step 2.6.12.5.6.3

Raise $\sqrt{3}$ to the power of $1$ .

$\csc (x) = - \frac{8 \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}}{\frac{6 \sqrt{2} + 2 \sqrt{6}}{3}}$

Step 2.6.12.5.6.4

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

$\csc (x) = - \frac{8 \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}}{\frac{6 \sqrt{2} + 2 \sqrt{6}}{3}}$

Step 2.6.12.5.6.5

Add $1$ and $1$ .

$\csc (x) = - \frac{8 \frac{\sqrt{3}}{{\sqrt{3}}^{2}}}{\frac{6 \sqrt{2} + 2 \sqrt{6}}{3}}$

Step 2.6.12.5.6.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.6.12.5.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\csc (x) = - \frac{8 \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}}{\frac{6 \sqrt{2} + 2 \sqrt{6}}{3}}$

Step 2.6.12.5.6.6.2

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

$\csc (x) = - \frac{8 \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}}{\frac{6 \sqrt{2} + 2 \sqrt{6}}{3}}$

Step 2.6.12.5.6.6.3

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

$\csc (x) = - \frac{8 \frac{\sqrt{3}}{3^{\frac{2}{2}}}}{\frac{6 \sqrt{2} + 2 \sqrt{6}}{3}}$

Step 2.6.12.5.6.6.4

Cancel the common factor of $2$ .

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Step 2.6.12.5.6.6.4.1

Cancel the common factor.

$\csc (x) = - \frac{8 \frac{\sqrt{3}}{3^{\frac{2}{2}}}}{\frac{6 \sqrt{2} + 2 \sqrt{6}}{3}}$

Step 2.6.12.5.6.6.4.2

Rewrite the expression.

$\csc (x) = - \frac{8 \frac{\sqrt{3}}{3^{1}}}{\frac{6 \sqrt{2} + 2 \sqrt{6}}{3}}$

Step 2.6.12.5.6.6.5

Evaluate the exponent.

$\csc (x) = - \frac{8 \frac{\sqrt{3}}{3}}{\frac{6 \sqrt{2} + 2 \sqrt{6}}{3}}$

Step 2.6.12.5.7

Combine $8$ and $\frac{\sqrt{3}}{3}$ .

$\csc (x) = - \frac{\frac{8 \sqrt{3}}{3}}{\frac{6 \sqrt{2} + 2 \sqrt{6}}{3}}$

Step 2.6.12.6

Multiply the numerator by the reciprocal of the denominator.

$\csc (x) = - (\frac{8 \sqrt{3}}{3} \cdot \frac{3}{6 \sqrt{2} + 2 \sqrt{6}})$

Step 2.6.12.7

Cancel the common factor of $3$ .

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Step 2.6.12.7.1

Cancel the common factor.

$\csc (x) = - (\frac{8 \sqrt{3}}{3} \cdot \frac{3}{6 \sqrt{2} + 2 \sqrt{6}})$

Step 2.6.12.7.2

Rewrite the expression.

$\csc (x) = - (8 \sqrt{3} \frac{1}{6 \sqrt{2} + 2 \sqrt{6}})$

Step 2.6.12.8

Combine $\frac{1}{6 \sqrt{2} + 2 \sqrt{6}}$ and $8$ .

$\csc (x) = - (\frac{8}{6 \sqrt{2} + 2 \sqrt{6}} \sqrt{3})$

Step 2.6.12.9

Combine $\frac{8}{6 \sqrt{2} + 2 \sqrt{6}}$ and $\sqrt{3}$ .

$\csc (x) = - \frac{8 \sqrt{3}}{6 \sqrt{2} + 2 \sqrt{6}}$

Step 2.6.12.10

Cancel the common factor of $8$ and $6 \sqrt{2} + 2 \sqrt{6}$ .

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Step 2.6.12.10.1

Factor $2$ out of $8 \sqrt{3}$ .

$\csc (x) = - \frac{2 (4 \sqrt{3})}{6 \sqrt{2} + 2 \sqrt{6}}$

Step 2.6.12.10.2

Cancel the common factors.

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Step 2.6.12.10.2.1

Factor $2$ out of $6 \sqrt{2}$ .

$\csc (x) = - \frac{2 (4 \sqrt{3})}{2 (3 \sqrt{2}) + 2 \sqrt{6}}$

Step 2.6.12.10.2.2

Factor $2$ out of $2 \sqrt{6}$ .

$\csc (x) = - \frac{2 (4 \sqrt{3})}{2 (3 \sqrt{2}) + 2 (\sqrt{6})}$

Step 2.6.12.10.2.3

Factor $2$ out of $2 (3 \sqrt{2}) + 2 (\sqrt{6})$ .

$\csc (x) = - \frac{2 (4 \sqrt{3})}{2 (3 \sqrt{2} + \sqrt{6})}$

Step 2.6.12.10.2.4

Cancel the common factor.

$\csc (x) = - \frac{2 (4 \sqrt{3})}{2 (3 \sqrt{2} + \sqrt{6})}$

Step 2.6.12.10.2.5

Rewrite the expression.

$\csc (x) = - \frac{4 \sqrt{3}}{3 \sqrt{2} + \sqrt{6}}$

Step 2.6.12.11

Multiply $\frac{4 \sqrt{3}}{3 \sqrt{2} + \sqrt{6}}$ by $\frac{3 \sqrt{2} - \sqrt{6}}{3 \sqrt{2} - \sqrt{6}}$ .

$\csc (x) = - (\frac{4 \sqrt{3}}{3 \sqrt{2} + \sqrt{6}} \cdot \frac{3 \sqrt{2} - \sqrt{6}}{3 \sqrt{2} - \sqrt{6}})$

Step 2.6.12.12

Multiply $\frac{4 \sqrt{3}}{3 \sqrt{2} + \sqrt{6}}$ by $\frac{3 \sqrt{2} - \sqrt{6}}{3 \sqrt{2} - \sqrt{6}}$ .

$\csc (x) = - \frac{4 \sqrt{3} (3 \sqrt{2} - \sqrt{6})}{(3 \sqrt{2} + \sqrt{6}) (3 \sqrt{2} - \sqrt{6})}$

Step 2.6.12.13

Expand the denominator using the FOIL method.

$\csc (x) = - \frac{4 \sqrt{3} (3 \sqrt{2} - \sqrt{6})}{9 {\sqrt{2}}^{2} - 3 \sqrt{12} + 3 \sqrt{12} - {\sqrt{6}}^{2}}$

Step 2.6.12.14

Simplify.

$\csc (x) = - \frac{4 \sqrt{3} (3 \sqrt{2} - \sqrt{6})}{12}$

Step 2.6.12.15

Cancel the common factor of $4$ and $12$ .

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Step 2.6.12.15.1

Factor $4$ out of $4 \sqrt{3} (3 \sqrt{2} - \sqrt{6})$ .

$\csc (x) = - \frac{4 (\sqrt{3} (3 \sqrt{2} - \sqrt{6}))}{12}$

Step 2.6.12.15.2

Cancel the common factors.

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Step 2.6.12.15.2.1

Factor $4$ out of $12$ .

$\csc (x) = - \frac{4 (\sqrt{3} (3 \sqrt{2} - \sqrt{6}))}{4 \cdot 3}$

Step 2.6.12.15.2.2

Cancel the common factor.

$\csc (x) = - \frac{4 (\sqrt{3} (3 \sqrt{2} - \sqrt{6}))}{4 \cdot 3}$

Step 2.6.12.15.2.3

Rewrite the expression.

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

Step 2.6.12.16

Apply the distributive property.

$\csc (x) = - \frac{\sqrt{3} (3 \sqrt{2}) + \sqrt{3} (- \sqrt{6})}{3}$

Step 2.6.12.17

Multiply $\sqrt{3} (3 \sqrt{2})$ .

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Step 2.6.12.17.1

Combine using the product rule for radicals.

$\csc (x) = - \frac{3 \sqrt{3 \cdot 2} + \sqrt{3} (- \sqrt{6})}{3}$

Step 2.6.12.17.2

Multiply $3$ by $2$ .

$\csc (x) = - \frac{3 \sqrt{6} + \sqrt{3} (- \sqrt{6})}{3}$

Step 2.6.12.18

Multiply $\sqrt{3} (- \sqrt{6})$ .

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Step 2.6.12.18.1

Combine using the product rule for radicals.

$\csc (x) = - \frac{3 \sqrt{6} - \sqrt{3 \cdot 6}}{3}$

Step 2.6.12.18.2

Multiply $3$ by $6$ .

$\csc (x) = - \frac{3 \sqrt{6} - \sqrt{18}}{3}$

Step 2.6.12.19

Simplify each term.

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Step 2.6.12.19.1

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 2.6.12.19.1.1

Factor $9$ out of $18$ .

$\csc (x) = - \frac{3 \sqrt{6} - \sqrt{9 (2)}}{3}$

Step 2.6.12.19.1.2

Rewrite $9$ as $3^{2}$ .

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

Step 2.6.12.19.2

Pull terms out from under the radical.

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

Step 2.6.12.19.3

Multiply $3$ by $- 1$ .

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

Step 2.6.12.20

Cancel the common factor of $3 \sqrt{6} - 3 \sqrt{2}$ and $3$ .

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Step 2.6.12.20.1

Factor $3$ out of $3 \sqrt{6}$ .

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

Step 2.6.12.20.2

Factor $3$ out of $- 3 \sqrt{2}$ .

$\csc (x) = - \frac{3 (\sqrt{6}) + 3 (- \sqrt{2})}{3}$

Step 2.6.12.20.3

Factor $3$ out of $3 (\sqrt{6}) + 3 (- \sqrt{2})$ .

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

Step 2.6.12.20.4

Cancel the common factors.

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Step 2.6.12.20.4.1

Factor $3$ out of $3$ .

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

Step 2.6.12.20.4.2

Cancel the common factor.

$\csc (x) = - \frac{3 (\sqrt{6} - \sqrt{2})}{3 \cdot 1}$

Step 2.6.12.20.4.3

Rewrite the expression.

$\csc (x) = - \frac{\sqrt{6} - \sqrt{2}}{1}$

Step 2.6.12.20.4.4

Divide $\sqrt{6} - \sqrt{2}$ by $1$ .

$\csc (x) = - (\sqrt{6} - \sqrt{2})$

Step 2.6.12.21

Apply the distributive property.

$\csc (x) = - \sqrt{6} - - \sqrt{2}$

Step 2.6.12.22

Multiply $- - \sqrt{2}$ .

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Step 2.6.12.22.1

Multiply $- 1$ by $- 1$ .

$\csc (x) = - \sqrt{6} + 1 \sqrt{2}$

Step 2.6.12.22.2

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

$\csc (x) = - \sqrt{6} + \sqrt{2}$

Step 3

Convert the right side of the equation to its decimal equivalent.

$\csc (x) = - 1.03527618$

Step 4

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

$x = arccsc (- 1.03527618)$

Step 5

Simplify the right side.

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Step 5.1

Evaluate $arccsc (- 1.03527618)$ .

$x = - \frac{5 π}{12}$

Step 6

The cosecant 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.

$x = 2 π + \frac{5 π}{12} + π$

Step 7

Simplify the expression to find the second solution.

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Step 7.1

Subtract $2 π$ from $2 π + \frac{5 π}{12} + π$ .

$x = 2 π + \frac{5 π}{12} + π - 2 π$

Step 7.2

The resulting angle of $\frac{17 π}{12}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{5 π}{12} + π$ .

$x = \frac{17 π}{12}$

Step 8

Find the period of $\csc (x)$ .

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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 $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 8.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 8.4

Divide $2 π$ by $1$ .

$2 π$

Step 9

Add $2 π$ to every negative angle to get positive angles.

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Step 9.1

Add $2 π$ to $- \frac{5 π}{12}$ to find the positive angle.

$- \frac{5 π}{12} + 2 π$

Step 9.2

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$2 π \cdot \frac{12}{12} - \frac{5 π}{12}$

Step 9.3

Combine fractions.

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Step 9.3.1

Combine $2 π$ and $\frac{12}{12}$ .

$\frac{2 π \cdot 12}{12} - \frac{5 π}{12}$

Step 9.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 12 - 5 π}{12}$

Step 9.4

Simplify the numerator.

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Step 9.4.1

Multiply $12$ by $2$ .

$\frac{24 π - 5 π}{12}$

Step 9.4.2

Subtract $5 π$ from $24 π$ .

$\frac{19 π}{12}$

Step 9.5

List the new angles.

$x = \frac{19 π}{12}$

Step 10

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

$x = \frac{17 π}{12} + 2 π n, \frac{19 π}{12} + 2 π n$ , for any integer $n$