Trigonometry Examples

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

Step 1

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

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

Step 2

Simplify both sides of the equation.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $\frac{2}{\sqrt{2}} (\frac{\sqrt{2}}{2} \csc (x - 1))$ .

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Step 2.1.1.1

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

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

Step 2.1.1.2

Combine and simplify the denominator.

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Step 2.1.1.2.1

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

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

Step 2.1.1.2.2

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

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

Step 2.1.1.2.3

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

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

Step 2.1.1.2.4

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

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

Step 2.1.1.2.5

Add $1$ and $1$ .

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

Step 2.1.1.2.6

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

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Step 2.1.1.2.6.1

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

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

Step 2.1.1.2.6.2

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

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

Step 2.1.1.2.6.3

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

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

Step 2.1.1.2.6.4

Cancel the common factor of $2$ .

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Step 2.1.1.2.6.4.1

Cancel the common factor.

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

Step 2.1.1.2.6.4.2

Rewrite the expression.

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

Step 2.1.1.2.6.5

Evaluate the exponent.

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

Step 2.1.1.3

Cancel the common factor of $2$ .

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Step 2.1.1.3.1

Cancel the common factor.

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

Step 2.1.1.3.2

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

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

Step 2.1.1.4

Combine $\frac{\sqrt{2}}{2}$ and $\csc (x - 1)$ .

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

Step 2.1.1.5

Multiply $\sqrt{2} \frac{\sqrt{2} \csc (x - 1)}{2}$ .

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Step 2.1.1.5.1

Combine $\sqrt{2}$ and $\frac{\sqrt{2} \csc (x - 1)}{2}$ .

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

Step 2.1.1.5.2

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

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

Step 2.1.1.5.3

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

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

Step 2.1.1.5.4

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

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

Step 2.1.1.5.5

Add $1$ and $1$ .

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

Step 2.1.1.6

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

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Step 2.1.1.6.1

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

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

Step 2.1.1.6.2

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

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

Step 2.1.1.6.3

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

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

Step 2.1.1.6.4

Cancel the common factor of $2$ .

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Step 2.1.1.6.4.1

Cancel the common factor.

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

Step 2.1.1.6.4.2

Rewrite the expression.

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

Step 2.1.1.6.5

Evaluate the exponent.

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

Step 2.1.1.7

Cancel the common factor of $2$ .

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Step 2.1.1.7.1

Cancel the common factor.

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

Step 2.1.1.7.2

Divide $\csc (x - 1)$ by $1$ .

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

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $\frac{2}{\sqrt{2}} \cdot 0$ .

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Step 2.2.1.1

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

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

Step 2.2.1.2

Combine and simplify the denominator.

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Step 2.2.1.2.1

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

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

Step 2.2.1.2.2

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

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

Step 2.2.1.2.3

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

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

Step 2.2.1.2.4

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

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

Step 2.2.1.2.5

Add $1$ and $1$ .

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

Step 2.2.1.2.6

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

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Step 2.2.1.2.6.1

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

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

Step 2.2.1.2.6.2

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

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

Step 2.2.1.2.6.3

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

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

Step 2.2.1.2.6.4

Cancel the common factor of $2$ .

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Step 2.2.1.2.6.4.1

Cancel the common factor.

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

Step 2.2.1.2.6.4.2

Rewrite the expression.

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

Step 2.2.1.2.6.5

Evaluate the exponent.

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

Step 2.2.1.3

Cancel the common factor of $2$ .

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Step 2.2.1.3.1

Cancel the common factor.

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

Step 2.2.1.3.2

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

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

Step 2.2.1.4

Multiply $\sqrt{2}$ by $0$ .

$\csc (x - 1) = 0$

Step 3

The range of cosecant is $y \leq - 1$ and $y \geq 1$ . Since $0$ does not fall in this range, there is no solution.

No solution