Trigonometry Examples

$(\frac{2 \sqrt{13}}{13}, - \frac{3 \sqrt{13}}{13})$

Step 1

To find the $\csc (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{2 \sqrt{13}}{13}, - \frac{3 \sqrt{13}}{13})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{2 \sqrt{13}}{13}, 0)$ , and $(\frac{2 \sqrt{13}}{13}, - \frac{3 \sqrt{13}}{13})$ .

Opposite : $- \frac{3 \sqrt{13}}{13}$

Adjacent : $\frac{2 \sqrt{13}}{13}$

Step 2

Find the hypotenuse using Pythagorean theorem $c = \sqrt{a^{2} + b^{2}}$ .

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Step 2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1.1

Apply the product rule to $\frac{2 \sqrt{13}}{13}$ .

$\sqrt{\frac{{(2 \sqrt{13})}^{2}}{13^{2}} + {(- \frac{3 \sqrt{13}}{13})}^{2}}$

Step 2.1.2

Apply the product rule to $2 \sqrt{13}$ .

$\sqrt{\frac{2^{2} {\sqrt{13}}^{2}}{13^{2}} + {(- \frac{3 \sqrt{13}}{13})}^{2}}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Raise $2$ to the power of $2$ .

$\sqrt{\frac{4 {\sqrt{13}}^{2}}{13^{2}} + {(- \frac{3 \sqrt{13}}{13})}^{2}}$

Step 2.2.2

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 2.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\sqrt{\frac{4 {(13^{\frac{1}{2}})}^{2}}{13^{2}} + {(- \frac{3 \sqrt{13}}{13})}^{2}}$

Step 2.2.2.2

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

$\sqrt{\frac{4 \cdot 13^{\frac{1}{2} \cdot 2}}{13^{2}} + {(- \frac{3 \sqrt{13}}{13})}^{2}}$

Step 2.2.2.3

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

$\sqrt{\frac{4 \cdot 13^{\frac{2}{2}}}{13^{2}} + {(- \frac{3 \sqrt{13}}{13})}^{2}}$

Step 2.2.2.4

Cancel the common factor of $2$ .

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Step 2.2.2.4.1

Cancel the common factor.

$\sqrt{\frac{4 \cdot 13^{\frac{2}{2}}}{13^{2}} + {(- \frac{3 \sqrt{13}}{13})}^{2}}$

Step 2.2.2.4.2

Rewrite the expression.

$\sqrt{\frac{4 \cdot 13^{1}}{13^{2}} + {(- \frac{3 \sqrt{13}}{13})}^{2}}$

Step 2.2.2.5

Evaluate the exponent.

$\sqrt{\frac{4 \cdot 13}{13^{2}} + {(- \frac{3 \sqrt{13}}{13})}^{2}}$

Step 2.3

Reduce the expression by cancelling the common factors.

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Step 2.3.1

Raise $13$ to the power of $2$ .

$\sqrt{\frac{4 \cdot 13}{169} + {(- \frac{3 \sqrt{13}}{13})}^{2}}$

Step 2.3.2

Multiply $4$ by $13$ .

$\sqrt{\frac{52}{169} + {(- \frac{3 \sqrt{13}}{13})}^{2}}$

Step 2.3.3

Cancel the common factor of $52$ and $169$ .

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Step 2.3.3.1

Factor $13$ out of $52$ .

$\sqrt{\frac{13 (4)}{169} + {(- \frac{3 \sqrt{13}}{13})}^{2}}$

Step 2.3.3.2

Cancel the common factors.

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Step 2.3.3.2.1

Factor $13$ out of $169$ .

$\sqrt{\frac{13 \cdot 4}{13 \cdot 13} + {(- \frac{3 \sqrt{13}}{13})}^{2}}$

Step 2.3.3.2.2

Cancel the common factor.

$\sqrt{\frac{13 \cdot 4}{13 \cdot 13} + {(- \frac{3 \sqrt{13}}{13})}^{2}}$

Step 2.3.3.2.3

Rewrite the expression.

$\sqrt{\frac{4}{13} + {(- \frac{3 \sqrt{13}}{13})}^{2}}$

Step 2.4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.4.1

Apply the product rule to $- \frac{3 \sqrt{13}}{13}$ .

$\sqrt{\frac{4}{13} + {(- 1)}^{2} {(\frac{3 \sqrt{13}}{13})}^{2}}$

Step 2.4.2

Apply the product rule to $\frac{3 \sqrt{13}}{13}$ .

$\sqrt{\frac{4}{13} + {(- 1)}^{2} \frac{{(3 \sqrt{13})}^{2}}{13^{2}}}$

Step 2.4.3

Apply the product rule to $3 \sqrt{13}$ .

$\sqrt{\frac{4}{13} + {(- 1)}^{2} \frac{3^{2} {\sqrt{13}}^{2}}{13^{2}}}$

Step 2.5

Simplify the expression.

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Step 2.5.1

Raise $- 1$ to the power of $2$ .

$\sqrt{\frac{4}{13} + 1 \frac{3^{2} {\sqrt{13}}^{2}}{13^{2}}}$

Step 2.5.2

Multiply $\frac{3^{2} {\sqrt{13}}^{2}}{13^{2}}$ by $1$ .

$\sqrt{\frac{4}{13} + \frac{3^{2} {\sqrt{13}}^{2}}{13^{2}}}$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Raise $3$ to the power of $2$ .

$\sqrt{\frac{4}{13} + \frac{9 {\sqrt{13}}^{2}}{13^{2}}}$

Step 2.6.2

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 2.6.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\sqrt{\frac{4}{13} + \frac{9 {(13^{\frac{1}{2}})}^{2}}{13^{2}}}$

Step 2.6.2.2

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

$\sqrt{\frac{4}{13} + \frac{9 \cdot 13^{\frac{1}{2} \cdot 2}}{13^{2}}}$

Step 2.6.2.3

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

$\sqrt{\frac{4}{13} + \frac{9 \cdot 13^{\frac{2}{2}}}{13^{2}}}$

Step 2.6.2.4

Cancel the common factor of $2$ .

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Step 2.6.2.4.1

Cancel the common factor.

$\sqrt{\frac{4}{13} + \frac{9 \cdot 13^{\frac{2}{2}}}{13^{2}}}$

Step 2.6.2.4.2

Rewrite the expression.

$\sqrt{\frac{4}{13} + \frac{9 \cdot 13^{1}}{13^{2}}}$

Step 2.6.2.5

Evaluate the exponent.

$\sqrt{\frac{4}{13} + \frac{9 \cdot 13}{13^{2}}}$

Step 2.7

Reduce the expression by cancelling the common factors.

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Step 2.7.1

Raise $13$ to the power of $2$ .

$\sqrt{\frac{4}{13} + \frac{9 \cdot 13}{169}}$

Step 2.7.2

Multiply $9$ by $13$ .

$\sqrt{\frac{4}{13} + \frac{117}{169}}$

Step 2.7.3

Cancel the common factor of $117$ and $169$ .

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Step 2.7.3.1

Factor $13$ out of $117$ .

$\sqrt{\frac{4}{13} + \frac{13 (9)}{169}}$

Step 2.7.3.2

Cancel the common factors.

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Step 2.7.3.2.1

Factor $13$ out of $169$ .

$\sqrt{\frac{4}{13} + \frac{13 \cdot 9}{13 \cdot 13}}$

Step 2.7.3.2.2

Cancel the common factor.

$\sqrt{\frac{4}{13} + \frac{13 \cdot 9}{13 \cdot 13}}$

Step 2.7.3.2.3

Rewrite the expression.

$\sqrt{\frac{4}{13} + \frac{9}{13}}$

Step 2.7.4

Simplify the expression.

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Step 2.7.4.1

Combine the numerators over the common denominator.

$\sqrt{\frac{4 + 9}{13}}$

Step 2.7.4.2

Add $4$ and $9$ .

$\sqrt{\frac{13}{13}}$

Step 2.7.4.3

Divide $13$ by $13$ .

$\sqrt{1}$

Step 2.7.4.4

Any root of $1$ is $1$ .

$1$

Step 3

$\csc (θ) = \frac{Hypotenuse}{Opposite}$ therefore $\csc (θ) = \frac{1}{- \frac{3 \sqrt{13}}{13}}$ .

$\frac{1}{- \frac{3 \sqrt{13}}{13}}$

Step 4

Simplify $\csc (θ)$ .

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Step 4.1

Cancel the common factor of $1$ and $- 1$ .

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Step 4.1.1

Rewrite $1$ as $- 1 (- 1)$ .

$\csc (θ) = \frac{- 1 \cdot - 1}{- \frac{3 \sqrt{13}}{13}}$

Step 4.1.2

Move the negative in front of the fraction.

$\csc (θ) = - \frac{1}{\frac{3 \sqrt{13}}{13}}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$\csc (θ) = - (1 (\frac{13}{3 \sqrt{13}}))$

Step 4.3

Multiply $\frac{13}{3 \sqrt{13}}$ by $1$ .

$\csc (θ) = - \frac{13}{3 \sqrt{13}}$

Step 4.4

Multiply $\frac{13}{3 \sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\csc (θ) = - (\frac{13}{3 \sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}})$

Step 4.5

Combine and simplify the denominator.

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Step 4.5.1

Multiply $\frac{13}{3 \sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\csc (θ) = - \frac{13 \sqrt{13}}{3 \sqrt{13} \sqrt{13}}$

Step 4.5.2

Move $\sqrt{13}$ .

$\csc (θ) = - \frac{13 \sqrt{13}}{3 (\sqrt{13} \sqrt{13})}$

Step 4.5.3

Raise $\sqrt{13}$ to the power of $1$ .

$\csc (θ) = - \frac{13 \sqrt{13}}{3 (\sqrt{13} \sqrt{13})}$

Step 4.5.4

Raise $\sqrt{13}$ to the power of $1$ .

$\csc (θ) = - \frac{13 \sqrt{13}}{3 (\sqrt{13} \sqrt{13})}$

Step 4.5.5

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

$\csc (θ) = - \frac{13 \sqrt{13}}{3 {\sqrt{13}}^{1 + 1}}$

Step 4.5.6

Add $1$ and $1$ .

$\csc (θ) = - \frac{13 \sqrt{13}}{3 {\sqrt{13}}^{2}}$

Step 4.5.7

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 4.5.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\csc (θ) = - \frac{13 \sqrt{13}}{3 {(13^{\frac{1}{2}})}^{2}}$

Step 4.5.7.2

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

$\csc (θ) = - \frac{13 \sqrt{13}}{3 \cdot 13^{\frac{1}{2} \cdot 2}}$

Step 4.5.7.3

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

$\csc (θ) = - \frac{13 \sqrt{13}}{3 \cdot 13^{\frac{2}{2}}}$

Step 4.5.7.4

Cancel the common factor of $2$ .

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Step 4.5.7.4.1

Cancel the common factor.

$\csc (θ) = - \frac{13 \sqrt{13}}{3 \cdot 13^{\frac{2}{2}}}$

Step 4.5.7.4.2

Rewrite the expression.

$\csc (θ) = - \frac{13 \sqrt{13}}{3 \cdot 13}$

Step 4.5.7.5

Evaluate the exponent.

$\csc (θ) = - \frac{13 \sqrt{13}}{3 \cdot 13}$

Step 4.6

Cancel the common factor of $13$ .

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Step 4.6.1

Cancel the common factor.

$\csc (θ) = - \frac{13 \sqrt{13}}{3 \cdot 13}$

Step 4.6.2

Rewrite the expression.

$\csc (θ) = - \frac{\sqrt{13}}{3}$

Step 5

Approximate the result.

$\csc (θ) = - \frac{\sqrt{13}}{3} \approx - 1.20185042$