Trigonometry Examples

$(\frac{5}{13}, - \frac{12}{13})$

Step 1

To find the $\csc (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{5}{13}, - \frac{12}{13})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{5}{13}, 0)$ , and $(\frac{5}{13}, - \frac{12}{13})$ .

Opposite : $- \frac{12}{13}$

Adjacent : $\frac{5}{13}$

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 $\frac{5}{13}$ .

$\sqrt{\frac{5^{2}}{13^{2}} + {(- \frac{12}{13})}^{2}}$

Step 2.2

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

$\sqrt{\frac{25}{13^{2}} + {(- \frac{12}{13})}^{2}}$

Step 2.3

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

$\sqrt{\frac{25}{169} + {(- \frac{12}{13})}^{2}}$

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 $- \frac{12}{13}$ .

$\sqrt{\frac{25}{169} + {(- 1)}^{2} {(\frac{12}{13})}^{2}}$

Step 2.4.2

Apply the product rule to $\frac{12}{13}$ .

$\sqrt{\frac{25}{169} + {(- 1)}^{2} \frac{12^{2}}{13^{2}}}$

Step 2.5

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

$\sqrt{\frac{25}{169} + 1 \frac{12^{2}}{13^{2}}}$

Step 2.6

Multiply $\frac{12^{2}}{13^{2}}$ by $1$ .

$\sqrt{\frac{25}{169} + \frac{12^{2}}{13^{2}}}$

Step 2.7

Raise $12$ to the power of $2$ .

$\sqrt{\frac{25}{169} + \frac{144}{13^{2}}}$

Step 2.8

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

$\sqrt{\frac{25}{169} + \frac{144}{169}}$

Step 2.9

Combine the numerators over the common denominator.

$\sqrt{\frac{25 + 144}{169}}$

Step 2.10

Add $25$ and $144$ .

$\sqrt{\frac{169}{169}}$

Step 2.11

Divide $169$ by $169$ .

$\sqrt{1}$

Step 2.12

Any root of $1$ is $1$ .

$1$

Step 3

$\csc (θ) = \frac{Hypotenuse}{Opposite}$ therefore $\csc (θ) = \frac{1}{- \frac{12}{13}}$ .

$\frac{1}{- \frac{12}{13}}$

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 $- 1 (- 1)$ .

$\csc (θ) = \frac{- 1 \cdot - 1}{- \frac{12}{13}}$

Step 4.1.2

Move the negative in front of the fraction.

$\csc (θ) = - \frac{1}{\frac{12}{13}}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$\csc (θ) = - (1 (\frac{13}{12}))$

Step 4.3

Multiply $\frac{13}{12}$ by $1$ .

$\csc (θ) = - \frac{13}{12}$

Step 5

Approximate the result.

$\csc (θ) = - \frac{13}{12} \approx - 1.08\overline{3}$