Trigonometry Examples

$(\frac{1}{8}, \frac{1}{6})$

Step 1

To find the $\cos (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{1}{8}, \frac{1}{6})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{1}{8}, 0)$ , and $(\frac{1}{8}, \frac{1}{6})$ .

Opposite : $\frac{1}{6}$

Adjacent : $\frac{1}{8}$

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 $\frac{1}{8}$ .

$\sqrt{\frac{1^{2}}{8^{2}} + {(\frac{1}{6})}^{2}}$

Step 2.2

One to any power is one.

$\sqrt{\frac{1}{8^{2}} + {(\frac{1}{6})}^{2}}$

Step 2.3

Raise $8$ to the power of $2$ .

$\sqrt{\frac{1}{64} + {(\frac{1}{6})}^{2}}$

Step 2.4

Apply the product rule to $\frac{1}{6}$ .

$\sqrt{\frac{1}{64} + \frac{1^{2}}{6^{2}}}$

Step 2.5

One to any power is one.

$\sqrt{\frac{1}{64} + \frac{1}{6^{2}}}$

Step 2.6

Raise $6$ to the power of $2$ .

$\sqrt{\frac{1}{64} + \frac{1}{36}}$

Step 2.7

To write $\frac{1}{64}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\sqrt{\frac{1}{64} \cdot \frac{9}{9} + \frac{1}{36}}$

Step 2.8

To write $\frac{1}{36}$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$\sqrt{\frac{1}{64} \cdot \frac{9}{9} + \frac{1}{36} \cdot \frac{16}{16}}$

Step 2.9

Write each expression with a common denominator of $576$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.9.1

Multiply $\frac{1}{64}$ by $\frac{9}{9}$ .

$\sqrt{\frac{9}{64 \cdot 9} + \frac{1}{36} \cdot \frac{16}{16}}$

Step 2.9.2

Multiply $64$ by $9$ .

$\sqrt{\frac{9}{576} + \frac{1}{36} \cdot \frac{16}{16}}$

Step 2.9.3

Multiply $\frac{1}{36}$ by $\frac{16}{16}$ .

$\sqrt{\frac{9}{576} + \frac{16}{36 \cdot 16}}$

Step 2.9.4

Multiply $36$ by $16$ .

$\sqrt{\frac{9}{576} + \frac{16}{576}}$

Step 2.10

Combine the numerators over the common denominator.

$\sqrt{\frac{9 + 16}{576}}$

Step 2.11

Add $9$ and $16$ .

$\sqrt{\frac{25}{576}}$

Step 2.12

Rewrite $\sqrt{\frac{25}{576}}$ as $\frac{\sqrt{25}}{\sqrt{576}}$ .

$\frac{\sqrt{25}}{\sqrt{576}}$

Step 2.13

Simplify the numerator.

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Step 2.13.1

Rewrite $25$ as $5^{2}$ .

$\frac{\sqrt{5^{2}}}{\sqrt{576}}$

Step 2.13.2

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

$\frac{5}{\sqrt{576}}$

Step 2.14

Simplify the denominator.

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Step 2.14.1

Rewrite $576$ as $24^{2}$ .

$\frac{5}{\sqrt{24^{2}}}$

Step 2.14.2

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

$\frac{5}{24}$

Step 3

$\cos (θ) = \frac{Adjacent}{Hypotenuse}$ therefore $\cos (θ) = \frac{\frac{1}{8}}{\frac{5}{24}}$ .

$\frac{\frac{1}{8}}{\frac{5}{24}}$

Step 4

Simplify $\cos (θ)$ .

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\cos (θ) = \frac{1}{8} \cdot \frac{24}{5}$

Step 4.2

Cancel the common factor of $8$ .

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Step 4.2.1

Factor $8$ out of $24$ .

$\cos (θ) = \frac{1}{8} \cdot \frac{8 (3)}{5}$

Step 4.2.2

Cancel the common factor.

$\cos (θ) = \frac{1}{8} \cdot \frac{8 \cdot 3}{5}$

Step 4.2.3

Rewrite the expression.

$\cos (θ) = \frac{3}{5}$

Step 5

Approximate the result.

$\cos (θ) = \frac{3}{5} \approx 0.6$