Trigonometry Examples

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

Step 1

To find the $\sin (θ)$ 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

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Combine the numerators over the common denominator.

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

Step 2.8

Add $25$ and $144$ .

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

Step 2.9

Divide $169$ by $169$ .

$\sqrt{1}$

Step 2.10

Any root of $1$ is $1$ .

$1$

Step 3

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

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

Step 4

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

$\sin (θ) = \frac{12}{13}$

Step 5

Approximate the result.

$\sin (θ) = \frac{12}{13} \approx 0.\overline{923076}$