Algebra Examples

$(\frac{5}{13}, y)$

Step 1

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{5}{13}, y)$ , draw the triangle between the three points $(0, 0)$ , $(\frac{5}{13}, 0)$ , and $(\frac{5}{13}, y)$ .

Opposite : $y$

Adjacent : $\frac{5}{13}$

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 $\frac{5}{13}$ .

$\sqrt{\frac{5^{2}}{13^{2}} + {(y)}^{2}}$

Step 2.2

Raise $5$ to the power of $2$ .

$\sqrt{\frac{25}{13^{2}} + {(y)}^{2}}$

Step 2.3

Raise $13$ to the power of $2$ .

$\sqrt{\frac{25}{169} + {(y)}^{2}}$

Step 2.4

To write $y^{2}$ as a fraction with a common denominator, multiply by $\frac{169}{169}$ .

$\sqrt{\frac{25}{169} + y^{2} \cdot \frac{169}{169}}$

Step 2.5

Simplify terms.

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Step 2.5.1

Combine $y^{2}$ and $\frac{169}{169}$ .

$\sqrt{\frac{25}{169} + \frac{y^{2} \cdot 169}{169}}$

Step 2.5.2

Combine the numerators over the common denominator.

$\sqrt{\frac{25 + y^{2} \cdot 169}{169}}$

Step 2.6

Move $169$ to the left of $y^{2}$ .

$\sqrt{\frac{25 + 169 y^{2}}{169}}$

Step 2.7

Rewrite $\frac{25 + 169 y^{2}}{169}$ as ${(\frac{1}{13})}^{2} (25 + 169 y^{2})$ .

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Step 2.7.1

Factor the perfect power $1^{2}$ out of $25 + 169 y^{2}$ .

$\sqrt{\frac{1^{2} (25 + 169 y^{2})}{169}}$

Step 2.7.2

Factor the perfect power $13^{2}$ out of $169$ .

$\sqrt{\frac{1^{2} (25 + 169 y^{2})}{13^{2} \cdot 1}}$

Step 2.7.3

Rearrange the fraction $\frac{1^{2} (25 + 169 y^{2})}{13^{2} \cdot 1}$ .

$\sqrt{{(\frac{1}{13})}^{2} (25 + 169 y^{2})}$

Step 2.8

Pull terms out from under the radical.

$\frac{1}{13} \sqrt{25 + 169 y^{2}}$

Step 2.9

Combine $\frac{1}{13}$ and $\sqrt{25 + 169 y^{2}}$ .

$\frac{\sqrt{25 + 169 y^{2}}}{13}$

Step 3

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{y}{\frac{\sqrt{25 + 169 y^{2}}}{13}}$ .

$\frac{y}{\frac{\sqrt{25 + 169 y^{2}}}{13}}$

Step 4

Simplify $\sin (θ)$ .

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\sin (θ) = y (\frac{13}{\sqrt{25 + 169 y^{2}}})$

Step 4.2

Multiply $\frac{13}{\sqrt{25 + 169 y^{2}}}$ by $\frac{\sqrt{25 + 169 y^{2}}}{\sqrt{25 + 169 y^{2}}}$ .

$\sin (θ) = y (\frac{13}{\sqrt{25 + 169 y^{2}}} \cdot \frac{\sqrt{25 + 169 y^{2}}}{\sqrt{25 + 169 y^{2}}})$

Step 4.3

Combine and simplify the denominator.

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Step 4.3.1

Multiply $\frac{13}{\sqrt{25 + 169 y^{2}}}$ by $\frac{\sqrt{25 + 169 y^{2}}}{\sqrt{25 + 169 y^{2}}}$ .

$\sin (θ) = y (\frac{13 \sqrt{25 + 169 y^{2}}}{\sqrt{25 + 169 y^{2}} \sqrt{25 + 169 y^{2}}})$

Step 4.3.2

Raise $\sqrt{25 + 169 y^{2}}$ to the power of $1$ .

$\sin (θ) = y (\frac{13 \sqrt{25 + 169 y^{2}}}{\sqrt{25 + 169 y^{2}} \sqrt{25 + 169 y^{2}}})$

Step 4.3.3

Raise $\sqrt{25 + 169 y^{2}}$ to the power of $1$ .

$\sin (θ) = y (\frac{13 \sqrt{25 + 169 y^{2}}}{\sqrt{25 + 169 y^{2}} \sqrt{25 + 169 y^{2}}})$

Step 4.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin (θ) = y (\frac{13 \sqrt{25 + 169 y^{2}}}{{\sqrt{25 + 169 y^{2}}}^{1 + 1}})$

Step 4.3.5

Add $1$ and $1$ .

$\sin (θ) = y (\frac{13 \sqrt{25 + 169 y^{2}}}{{\sqrt{25 + 169 y^{2}}}^{2}})$

Step 4.3.6

Rewrite ${\sqrt{25 + 169 y^{2}}}^{2}$ as $25 + 169 y^{2}$ .

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Step 4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{25 + 169 y^{2}}$ as ${(25 + 169 y^{2})}^{\frac{1}{2}}$ .

$\sin (θ) = y (\frac{13 \sqrt{25 + 169 y^{2}}}{{({(25 + 169 y^{2})}^{\frac{1}{2}})}^{2}})$

Step 4.3.6.2

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

$\sin (θ) = y (\frac{13 \sqrt{25 + 169 y^{2}}}{{(25 + 169 y^{2})}^{\frac{1}{2} \cdot 2}})$

Step 4.3.6.3

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

$\sin (θ) = y (\frac{13 \sqrt{25 + 169 y^{2}}}{{(25 + 169 y^{2})}^{\frac{2}{2}}})$

Step 4.3.6.4

Cancel the common factor of $2$ .

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Step 4.3.6.4.1

Cancel the common factor.

$\sin (θ) = y (\frac{13 \sqrt{25 + 169 y^{2}}}{{(25 + 169 y^{2})}^{\frac{2}{2}}})$

Step 4.3.6.4.2

Rewrite the expression.

$\sin (θ) = y (\frac{13 \sqrt{25 + 169 y^{2}}}{25 + 169 y^{2}})$

Step 4.3.6.5

Simplify.

$\sin (θ) = y (\frac{13 \sqrt{25 + 169 y^{2}}}{25 + 169 y^{2}})$

Step 4.4

Combine $y$ and $\frac{13 \sqrt{25 + 169 y^{2}}}{25 + 169 y^{2}}$ .

$\sin (θ) = \frac{y (13 \sqrt{25 + 169 y^{2}})}{25 + 169 y^{2}}$

Step 4.5

Move $13$ to the left of $y$ .

$\sin (θ) = \frac{13 y \sqrt{25 + 169 y^{2}}}{25 + 169 y^{2}}$

Step 5

Approximate the result.

$\sin (θ) = \frac{13 y \sqrt{25 + 169 y^{2}}}{25 + 169 y^{2}} \approx \frac{13 y \sqrt{25 + 169 y^{2}}}{25 + 169 y^{2}}$