Trigonometry Examples

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

Step 1

Combine $\frac{\sqrt{2}}{2}$ and $\csc (x)$ .

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

Step 2

Add $1$ to both sides of the equation.

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

Step 3

Multiply both sides of the equation by $\frac{2}{\sqrt{2}}$ .

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

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify $\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2} \csc (x)}{2}$ .

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Step 4.1.1.1

Cancel the common factor of $2$ .

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Step 4.1.1.1.1

Cancel the common factor.

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

Step 4.1.1.1.2

Rewrite the expression.

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

Step 4.1.1.2

Cancel the common factor of $\sqrt{2}$ .

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Step 4.1.1.2.1

Factor $\sqrt{2}$ out of $\sqrt{2} \csc (x)$ .

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

Step 4.1.1.2.2

Cancel the common factor.

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

Step 4.1.1.2.3

Rewrite the expression.

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

Step 4.2

Simplify the right side.

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Step 4.2.1

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

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Step 4.2.1.1

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

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

Step 4.2.1.2

Combine and simplify the denominator.

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Step 4.2.1.2.1

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

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

Step 4.2.1.2.2

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

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

Step 4.2.1.2.3

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

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

Step 4.2.1.2.4

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

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

Step 4.2.1.2.5

Add $1$ and $1$ .

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

Step 4.2.1.2.6

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

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Step 4.2.1.2.6.1

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

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

Step 4.2.1.2.6.2

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

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

Step 4.2.1.2.6.3

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

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

Step 4.2.1.2.6.4

Cancel the common factor of $2$ .

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Step 4.2.1.2.6.4.1

Cancel the common factor.

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

Step 4.2.1.2.6.4.2

Rewrite the expression.

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

Step 4.2.1.2.6.5

Evaluate the exponent.

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

Step 4.2.1.3

Cancel the common factor of $2$ .

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Step 4.2.1.3.1

Cancel the common factor.

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

Step 4.2.1.3.2

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

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

Step 4.2.1.4

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

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

Step 5

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

$x = arccsc (\sqrt{2})$

Step 6

Simplify the right side.

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Step 6.1

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

$x = \frac{π}{4}$

Step 7

The cosecant 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.

$x = π - \frac{π}{4}$

Step 8

Simplify $π - \frac{π}{4}$ .

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Step 8.1

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

$x = π \cdot \frac{4}{4} - \frac{π}{4}$

Step 8.2

Combine fractions.

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Step 8.2.1

Combine $π$ and $\frac{4}{4}$ .

$x = \frac{π \cdot 4}{4} - \frac{π}{4}$

Step 8.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 4 - π}{4}$

Step 8.3

Simplify the numerator.

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Step 8.3.1

Move $4$ to the left of $π$ .

$x = \frac{4 \cdot π - π}{4}$

Step 8.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{4}$

Step 9

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

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Step 9.1

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

$\frac{2 π}{| b |}$

Step 9.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 9.3

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

$\frac{2 π}{1}$

Step 9.4

Divide $2 π$ by $1$ .

$2 π$

Step 10

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

$x = \frac{π}{4} + 2 π n, \frac{3 π}{4} + 2 π n$ , for any integer $n$