Find the Sine (3,2)

$(3, 2)$

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(3, 2)$ , draw the triangle between the three points $(0, 0)$ , $(3, 0)$ , and $(3, 2)$ .

Opposite : $2$

Adjacent : $3$

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

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Raise $3$ to the power of $2$ .

$\sqrt{9 + {(2)}^{2}}$

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

$\sqrt{9 + 4}$

Add $9$ and $4$ .

$\sqrt{13}$

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{2}{\sqrt{13}}$ .

$\frac{2}{\sqrt{13}}$

Simplify $\sin (θ)$ .

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Multiply $\frac{2}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\sin (θ) = \frac{2}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}}$

Combine and simplify the denominator.

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Multiply $\frac{2}{\sqrt{13}}$ and $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\sin (θ) = \frac{2 \sqrt{13}}{\sqrt{13} \sqrt{13}}$

Raise $\sqrt{13}$ to the power of $1$ .

$\sin (θ) = \frac{2 \sqrt{13}}{\sqrt{13} \sqrt{13}}$

Raise $\sqrt{13}$ to the power of $1$ .

$\sin (θ) = \frac{2 \sqrt{13}}{\sqrt{13} \sqrt{13}}$

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

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

Add $1$ and $1$ .

$\sin (θ) = \frac{2 \sqrt{13}}{{\sqrt{13}}^{2}}$

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\sin (θ) = \frac{2 \sqrt{13}}{{(13^{\frac{1}{2}})}^{2}}$

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

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

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

$\sin (θ) = \frac{2 \sqrt{13}}{13^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\sin (θ) = \frac{2 \sqrt{13}}{13^{\frac{2}{2}}}$

Divide $1$ by $1$ .

$\sin (θ) = \frac{2 \sqrt{13}}{13}$

Evaluate the exponent.

$\sin (θ) = \frac{2 \sqrt{13}}{13}$

Approximate the result.

$\sin (θ) = \frac{2 \sqrt{13}}{13} \approx 0.55470019$

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