Precalculus Examples

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

Step 1

To find the $\sin (θ)$ 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

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1.1

Apply the product rule to $- \frac{1}{8}$ .

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

Step 2.1.2

Apply the product rule to $\frac{1}{8}$ .

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

Step 2.2

Raise $- 1$ to the power of $2$ .

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

Step 2.3

Multiply $\frac{1^{2}}{8^{2}}$ by $1$ .

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

Step 2.4

One to any power is one.

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

One to any power is one.

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

Step 2.8

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

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

Step 2.9

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 2.10

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 2.11

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

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Step 2.11.1

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

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

Step 2.11.2

Multiply $64$ by $9$ .

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

Step 2.11.3

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

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

Step 2.11.4

Multiply $36$ by $16$ .

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

Step 2.12

Combine the numerators over the common denominator.

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

Step 2.13

Add $9$ and $16$ .

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

Step 2.14

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

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

Step 2.15

Simplify the numerator.

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Step 2.15.1

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

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

Step 2.15.2

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

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

Step 2.16

Simplify the denominator.

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Step 2.16.1

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

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

Step 2.16.2

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

$\frac{5}{24}$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{\frac{1}{6}}{\frac{5}{24}}$ .

$\frac{\frac{1}{6}}{\frac{5}{24}}$

Step 4

Simplify $\sin (θ)$ .

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\sin (θ) = \frac{1}{6} \cdot \frac{24}{5}$

Step 4.2

Cancel the common factor of $6$ .

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Step 4.2.1

Factor $6$ out of $24$ .

$\sin (θ) = \frac{1}{6} \cdot \frac{6 (4)}{5}$

Step 4.2.2

Cancel the common factor.

$\sin (θ) = \frac{1}{6} \cdot \frac{6 \cdot 4}{5}$

Step 4.2.3

Rewrite the expression.

$\sin (θ) = \frac{4}{5}$

Step 5

Approximate the result.

$\sin (θ) = \frac{4}{5} \approx 0.8$