Trigonometry Examples

(- 5, - 12)

$(- 5, - 12)$

Step 1

To find the

\sin (θ)

$\sin (θ)$ between the x-axis and the line between the points

(0, 0)

$(0, 0)$ and

(- 5, - 12)

$(- 5, - 12)$ , draw the triangle between the three points

(0, 0)

$(0, 0)$ ,

(- 5, 0)

$(- 5, 0)$ , and

(- 5, - 12)

$(- 5, - 12)$ .

Opposite :

- 12

$- 12$

Adjacent :

- 5

$- 5$

Step 2

Find the hypotenuse using Pythagorean theorem

c = \sqrt{a^{2} + b^{2}}

$c = \sqrt{a^{2} + b^{2}}$ .

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Step 2.1

Raise

- 5

$- 5$ to the power of

2

$2$ .

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

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

Step 2.2

Raise

- 12

$- 12$ to the power of

2

$2$ .

\sqrt{25 + 144}

$\sqrt{25 + 144}$

Step 2.3

Add

25

$25$ and

144

$144$ .

\sqrt{169}

$\sqrt{169}$

Step 2.4

Rewrite

169

$169$ as

13^{2}

$13^{2}$ .

\sqrt{13^{2}}

$\sqrt{13^{2}}$

Step 2.5

Pull terms out from under the radical, assuming positive real numbers.

13

$13$

13

$13$

Step 3

\sin (θ) = \frac{Opposite}{Hypotenuse}

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore

\sin (θ) = \frac{- 12}{13}

$\sin (θ) = \frac{- 12}{13}$ .

\frac{- 12}{13}

$\frac{- 12}{13}$

Step 4

Move the negative in front of the fraction.

\sin (θ) = - \frac{12}{13}

$\sin (θ) = - \frac{12}{13}$

Step 5

Approximate the result.

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

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