Trigonometry Examples

$(\frac{1}{5}, - \frac{2 \sqrt{6}}{5})$

Step 1

To find the $\csc (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{1}{5}, - \frac{2 \sqrt{6}}{5})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{1}{5}, 0)$ , and $(\frac{1}{5}, - \frac{2 \sqrt{6}}{5})$ .

Opposite : $- \frac{2 \sqrt{6}}{5}$

Adjacent : $\frac{1}{5}$

Step 2

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

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Step 2.1

Apply the product rule to $\frac{1}{5}$ .

$\sqrt{\frac{1^{2}}{5^{2}} + {(- \frac{2 \sqrt{6}}{5})}^{2}}$

Step 2.2

One to any power is one.

$\sqrt{\frac{1}{5^{2}} + {(- \frac{2 \sqrt{6}}{5})}^{2}}$

Step 2.3

Raise $5$ to the power of $2$ .

$\sqrt{\frac{1}{25} + {(- \frac{2 \sqrt{6}}{5})}^{2}}$

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 $- \frac{2 \sqrt{6}}{5}$ .

$\sqrt{\frac{1}{25} + {(- 1)}^{2} {(\frac{2 \sqrt{6}}{5})}^{2}}$

Step 2.4.2

Apply the product rule to $\frac{2 \sqrt{6}}{5}$ .

$\sqrt{\frac{1}{25} + {(- 1)}^{2} \frac{{(2 \sqrt{6})}^{2}}{5^{2}}}$

Step 2.4.3

Apply the product rule to $2 \sqrt{6}$ .

$\sqrt{\frac{1}{25} + {(- 1)}^{2} \frac{2^{2} {\sqrt{6}}^{2}}{5^{2}}}$

Step 2.5

Simplify the expression.

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Step 2.5.1

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

$\sqrt{\frac{1}{25} + 1 \frac{2^{2} {\sqrt{6}}^{2}}{5^{2}}}$

Step 2.5.2

Multiply $\frac{2^{2} {\sqrt{6}}^{2}}{5^{2}}$ by $1$ .

$\sqrt{\frac{1}{25} + \frac{2^{2} {\sqrt{6}}^{2}}{5^{2}}}$

Step 2.6

Simplify the numerator.

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Step 2.6.1

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

$\sqrt{\frac{1}{25} + \frac{4 {\sqrt{6}}^{2}}{5^{2}}}$

Step 2.6.2

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 2.6.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\sqrt{\frac{1}{25} + \frac{4 {(6^{\frac{1}{2}})}^{2}}{5^{2}}}$

Step 2.6.2.2

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

$\sqrt{\frac{1}{25} + \frac{4 \cdot 6^{\frac{1}{2} \cdot 2}}{5^{2}}}$

Step 2.6.2.3

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

$\sqrt{\frac{1}{25} + \frac{4 \cdot 6^{\frac{2}{2}}}{5^{2}}}$

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.

$\sqrt{\frac{1}{25} + \frac{4 \cdot 6^{\frac{2}{2}}}{5^{2}}}$

Step 2.6.2.4.2

Rewrite the expression.

$\sqrt{\frac{1}{25} + \frac{4 \cdot 6^{1}}{5^{2}}}$

Step 2.6.2.5

Evaluate the exponent.

$\sqrt{\frac{1}{25} + \frac{4 \cdot 6}{5^{2}}}$

Step 2.7

Simplify the expression.

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Step 2.7.1

Raise $5$ to the power of $2$ .

$\sqrt{\frac{1}{25} + \frac{4 \cdot 6}{25}}$

Step 2.7.2

Multiply $4$ by $6$ .

$\sqrt{\frac{1}{25} + \frac{24}{25}}$

Step 2.7.3

Combine the numerators over the common denominator.

$\sqrt{\frac{1 + 24}{25}}$

Step 2.7.4

Add $1$ and $24$ .

$\sqrt{\frac{25}{25}}$

Step 2.7.5

Divide $25$ by $25$ .

$\sqrt{1}$

Step 2.7.6

Any root of $1$ is $1$ .

$1$

Step 3

$\csc (θ) = \frac{Hypotenuse}{Opposite}$ therefore $\csc (θ) = \frac{1}{- \frac{2 \sqrt{6}}{5}}$ .

$\frac{1}{- \frac{2 \sqrt{6}}{5}}$

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 $- 1 (- 1)$ .

$\csc (θ) = \frac{- 1 \cdot - 1}{- \frac{2 \sqrt{6}}{5}}$

Step 4.1.2

Move the negative in front of the fraction.

$\csc (θ) = - \frac{1}{\frac{2 \sqrt{6}}{5}}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$\csc (θ) = - (1 (\frac{5}{2 \sqrt{6}}))$

Step 4.3

Multiply $\frac{5}{2 \sqrt{6}}$ by $1$ .

$\csc (θ) = - \frac{5}{2 \sqrt{6}}$

Step 4.4

Multiply $\frac{5}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\csc (θ) = - (\frac{5}{2 \sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}})$

Step 4.5

Combine and simplify the denominator.

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Step 4.5.1

Multiply $\frac{5}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\csc (θ) = - \frac{5 \sqrt{6}}{2 \sqrt{6} \sqrt{6}}$

Step 4.5.2

Move $\sqrt{6}$ .

$\csc (θ) = - \frac{5 \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

Step 4.5.3

Raise $\sqrt{6}$ to the power of $1$ .

$\csc (θ) = - \frac{5 \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

Step 4.5.4

Raise $\sqrt{6}$ to the power of $1$ .

$\csc (θ) = - \frac{5 \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

Step 4.5.5

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

$\csc (θ) = - \frac{5 \sqrt{6}}{2 {\sqrt{6}}^{1 + 1}}$

Step 4.5.6

Add $1$ and $1$ .

$\csc (θ) = - \frac{5 \sqrt{6}}{2 {\sqrt{6}}^{2}}$

Step 4.5.7

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 4.5.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\csc (θ) = - \frac{5 \sqrt{6}}{2 {(6^{\frac{1}{2}})}^{2}}$

Step 4.5.7.2

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

$\csc (θ) = - \frac{5 \sqrt{6}}{2 \cdot 6^{\frac{1}{2} \cdot 2}}$

Step 4.5.7.3

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

$\csc (θ) = - \frac{5 \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

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.

$\csc (θ) = - \frac{5 \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

Step 4.5.7.4.2

Rewrite the expression.

$\csc (θ) = - \frac{5 \sqrt{6}}{2 \cdot 6}$

Step 4.5.7.5

Evaluate the exponent.

$\csc (θ) = - \frac{5 \sqrt{6}}{2 \cdot 6}$

Step 4.6

Multiply $2$ by $6$ .

$\csc (θ) = - \frac{5 \sqrt{6}}{12}$

Step 5

Approximate the result.

$\csc (θ) = - \frac{5 \sqrt{6}}{12} \approx - 1.02062072$