Trigonometry Examples

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

Step 1

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

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

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

Step 2

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

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Step 2.1

Move the negative in front of the fraction.

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

Step 2.2

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

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Step 2.2.1

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

$\sqrt{{(- 1)}^{2} {(\frac{12}{13})}^{2} + {(\frac{5}{13})}^{2}}$

Step 2.2.2

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

$\sqrt{{(- 1)}^{2} \frac{12^{2}}{13^{2}} + {(\frac{5}{13})}^{2}}$

Step 2.3

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

$\sqrt{1 \frac{12^{2}}{13^{2}} + {(\frac{5}{13})}^{2}}$

Step 2.4

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

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

Step 2.5

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

$\sqrt{\frac{144}{13^{2}} + {(\frac{5}{13})}^{2}}$

Step 2.6

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

$\sqrt{\frac{144}{169} + {(\frac{5}{13})}^{2}}$

Step 2.7

Apply the product rule to $\frac{5}{13}$ .

$\sqrt{\frac{144}{169} + \frac{5^{2}}{13^{2}}}$

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

Combine the numerators over the common denominator.

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

Step 2.11

Add $144$ and $25$ .

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

Step 2.12

Divide $169$ by $169$ .

$\sqrt{1}$

Step 2.13

Any root of $1$ is $1$ .

$1$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{\frac{5}{13}}{1}$ .

$\frac{\frac{5}{13}}{1}$

Step 4

Divide $\frac{5}{13}$ by $1$ .

$\sin (θ) = \frac{5}{13}$

Step 5

Approximate the result.

$\sin (θ) = \frac{5}{13} \approx 0.\overline{384615}$