Precalculus Examples

$(- \frac{\sqrt{65}}{9}, - \frac{4}{9})$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- \frac{\sqrt{65}}{9}, - \frac{4}{9})$ , draw the triangle between the three points $(0, 0)$ , $(- \frac{\sqrt{65}}{9}, 0)$ , and $(- \frac{\sqrt{65}}{9}, - \frac{4}{9})$ .

Opposite : $- \frac{4}{9}$

Adjacent : $- \frac{\sqrt{65}}{9}$

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 $- \frac{\sqrt{65}}{9}$ .

$\sqrt{{(- 1)}^{2} {(\frac{\sqrt{65}}{9})}^{2} + {(- \frac{4}{9})}^{2}}$

Step 2.1.2

Apply the product rule to $\frac{\sqrt{65}}{9}$ .

$\sqrt{{(- 1)}^{2} \frac{{\sqrt{65}}^{2}}{9^{2}} + {(- \frac{4}{9})}^{2}}$

Step 2.2

Simplify the expression.

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Step 2.2.1

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

$\sqrt{1 \frac{{\sqrt{65}}^{2}}{9^{2}} + {(- \frac{4}{9})}^{2}}$

Step 2.2.2

Multiply $\frac{{\sqrt{65}}^{2}}{9^{2}}$ by $1$ .

$\sqrt{\frac{{\sqrt{65}}^{2}}{9^{2}} + {(- \frac{4}{9})}^{2}}$

Step 2.3

Rewrite ${\sqrt{65}}^{2}$ as $65$ .

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Step 2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{65}$ as $65^{\frac{1}{2}}$ .

$\sqrt{\frac{{(65^{\frac{1}{2}})}^{2}}{9^{2}} + {(- \frac{4}{9})}^{2}}$

Step 2.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{\frac{65^{\frac{1}{2} \cdot 2}}{9^{2}} + {(- \frac{4}{9})}^{2}}$

Step 2.3.3

Combine $\frac{1}{2}$ and $2$ .

$\sqrt{\frac{65^{\frac{2}{2}}}{9^{2}} + {(- \frac{4}{9})}^{2}}$

Step 2.3.4

Cancel the common factor of $2$ .

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Step 2.3.4.1

Cancel the common factor.

$\sqrt{\frac{65^{\frac{2}{2}}}{9^{2}} + {(- \frac{4}{9})}^{2}}$

Step 2.3.4.2

Rewrite the expression.

$\sqrt{\frac{65^{1}}{9^{2}} + {(- \frac{4}{9})}^{2}}$

Step 2.3.5

Evaluate the exponent.

$\sqrt{\frac{65}{9^{2}} + {(- \frac{4}{9})}^{2}}$

Step 2.4

Raise $9$ to the power of $2$ .

$\sqrt{\frac{65}{81} + {(- \frac{4}{9})}^{2}}$

Step 2.5

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

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Step 2.5.1

Apply the product rule to $- \frac{4}{9}$ .

$\sqrt{\frac{65}{81} + {(- 1)}^{2} {(\frac{4}{9})}^{2}}$

Step 2.5.2

Apply the product rule to $\frac{4}{9}$ .

$\sqrt{\frac{65}{81} + {(- 1)}^{2} \frac{4^{2}}{9^{2}}}$

Step 2.6

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

$\sqrt{\frac{65}{81} + 1 \frac{4^{2}}{9^{2}}}$

Step 2.7

Multiply $\frac{4^{2}}{9^{2}}$ by $1$ .

$\sqrt{\frac{65}{81} + \frac{4^{2}}{9^{2}}}$

Step 2.8

Raise $4$ to the power of $2$ .

$\sqrt{\frac{65}{81} + \frac{16}{9^{2}}}$

Step 2.9

Raise $9$ to the power of $2$ .

$\sqrt{\frac{65}{81} + \frac{16}{81}}$

Step 2.10

Combine the numerators over the common denominator.

$\sqrt{\frac{65 + 16}{81}}$

Step 2.11

Add $65$ and $16$ .

$\sqrt{\frac{81}{81}}$

Step 2.12

Divide $81$ by $81$ .

$\sqrt{1}$

Step 2.13

Any root of $1$ is $1$ .

$1$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{- \frac{4}{9}}{1}$ .

$\frac{- \frac{4}{9}}{1}$

Step 4

Divide $- \frac{4}{9}$ by $1$ .

$\sin (θ) = - \frac{4}{9}$

Step 5

Approximate the result.

$\sin (θ) = - \frac{4}{9} \approx - 0.\overline{4}$