Trigonometry Examples

$(16, - 12)$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(16, - 12)$ , draw the triangle between the three points $(0, 0)$ , $(16, 0)$ , and $(16, - 12)$ .

Opposite : $- 12$

Adjacent : $16$

Step 2

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

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Step 2.1

Raise $16$ to the power of $2$ .

$\sqrt{256 + {(- 12)}^{2}}$

Step 2.2

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

$\sqrt{256 + 144}$

Step 2.3

Add $256$ and $144$ .

$\sqrt{400}$

Step 2.4

Rewrite $400$ as $20^{2}$ .

$\sqrt{20^{2}}$

Step 2.5

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

$20$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{16}{20}$ .

$\frac{16}{20}$

Step 4

Cancel the common factor of $16$ and $20$ .

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Step 4.1

Factor $4$ out of $16$ .

$\cos (θ) = \frac{4 (4)}{20}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $4$ out of $20$ .

$\cos (θ) = \frac{4 \cdot 4}{4 \cdot 5}$

Step 4.2.2

Cancel the common factor.

$\cos (θ) = \frac{4 \cdot 4}{4 \cdot 5}$

Step 4.2.3

Rewrite the expression.

$\cos (θ) = \frac{4}{5}$

Step 5

Approximate the result.

$\cos (θ) = \frac{4}{5} \approx 0.8$