Trigonometry Examples

(- 15, 36)

$(- 15, 36)$

Step 1

To find the

sin (θ)

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

(0, 0)

$(0, 0)$ and

(- 15, 36)

$(- 15, 36)$ , draw the triangle between the three points

(0, 0)

$(0, 0)$ ,

(- 15, 0)

$(- 15, 0)$ , and

(- 15, 36)

$(- 15, 36)$ .

Opposite :

36

$36$

Adjacent :

- 15

$- 15$

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

- 15

$- 15$ to the power of

2

$2$ .

\sqrt{225 + {(36)}^{2}}

$\sqrt{225 + {(36)}^{2}}$

Step 2.2

Raise

36

$36$ to the power of

2

$2$ .

\sqrt{225 + 1296}

$\sqrt{225 + 1296}$

Step 2.3

Add

225

$225$ and

1296

$1296$ .

\sqrt{1521}

$\sqrt{1521}$

Step 2.4

Rewrite

1521

$1521$ as

39^{2}

$39^{2}$ .

\sqrt{39^{2}}

$\sqrt{39^{2}}$

Step 2.5

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

39

$39$

39

$39$

Step 3

sin (θ) = \frac{Opposite}{Hypotenuse}

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

sin (θ) = \frac{36}{39}

$\sin (θ) = \frac{36}{39}$ .

\frac{36}{39}

$\frac{36}{39}$

Step 4

Cancel the common factor of

36

$36$ and

39

$39$ .

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Step 4.1

Factor

3

$3$ out of

36

$36$ .

sin (θ) = \frac{3 (12)}{39}

$\sin (θ) = \frac{3 (12)}{39}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor

3

$3$ out of

39

$39$ .

sin (θ) = \frac{3 \cdot 12}{3 \cdot 13}

$\sin (θ) = \frac{3 \cdot 12}{3 \cdot 13}$

Step 4.2.2

Cancel the common factor.

$\sin (θ) = \frac{3 \cdot 12}{3 \cdot 13}$

Step 4.2.3

Rewrite the expression.

$\sin (θ) = \frac{12}{13}$

Step 5

Approximate the result.

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