Trigonometry Examples

(- 12, 9)

$(- 12, 9)$

Step 1

To find the

\sin (θ)

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

(0, 0)

$(0, 0)$ and

(- 12, 9)

$(- 12, 9)$ , draw the triangle between the three points

(0, 0)

$(0, 0)$ ,

(- 12, 0)

$(- 12, 0)$ , and

(- 12, 9)

$(- 12, 9)$ .

Opposite :

9

$9$

Adjacent :

- 12

$- 12$

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

- 12

$- 12$ to the power of

2

$2$ .

\sqrt{144 + {(9)}^{2}}

$\sqrt{144 + {(9)}^{2}}$

Step 2.2

Raise

9

$9$ to the power of

2

$2$ .

\sqrt{144 + 81}

$\sqrt{144 + 81}$

Step 2.3

Add

144

$144$ and

81

$81$ .

\sqrt{225}

$\sqrt{225}$

Step 2.4

Rewrite

225

$225$ as

15^{2}

$15^{2}$ .

\sqrt{15^{2}}

$\sqrt{15^{2}}$

Step 2.5

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

15

$15$

15

$15$

Step 3

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

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

\sin (θ) = \frac{9}{15}

$\sin (θ) = \frac{9}{15}$ .

\frac{9}{15}

$\frac{9}{15}$

Step 4

Cancel the common factor of

9

$9$ and

15

$15$ .

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Step 4.1

Factor

3

$3$ out of

9

$9$ .

\sin (θ) = \frac{3 (3)}{15}

$\sin (θ) = \frac{3 (3)}{15}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor

3

$3$ out of

15

$15$ .

\sin (θ) = \frac{3 \cdot 3}{3 \cdot 5}

$\sin (θ) = \frac{3 \cdot 3}{3 \cdot 5}$

Step 4.2.2

Cancel the common factor.

\sin (θ) = \frac{3 \cdot 3}{3 \cdot 5}

$\sin (θ) = \frac{3 \cdot 3}{3 \cdot 5}$

Step 4.2.3

Rewrite the expression.

\sin (θ) = \frac{3}{5}

$\sin (θ) = \frac{3}{5}$

\sin (θ) = \frac{3}{5}

$\sin (θ) = \frac{3}{5}$

\sin (θ) = \frac{3}{5}

$\sin (θ) = \frac{3}{5}$

Step 5

Approximate the result.

\sin (θ) = \frac{3}{5} \approx 0.6

$\sin (θ) = \frac{3}{5} \approx 0.6$