Find the Sine (-6, square root of 3)

$(- 6, \sqrt{3})$

To find the $\sin (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- 6, \sqrt{3})$ , draw the triangle between the three points $(0, 0)$ , $(- 6, 0)$ , and $(- 6, \sqrt{3})$ .

Opposite : $\sqrt{3}$

Adjacent : $- 6$

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

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

$\sqrt{36 + {(\sqrt{3})}^{2}}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\sqrt{36 + {(3^{\frac{1}{2}})}^{2}}$

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

$\sqrt{36 + 3^{\frac{1}{2} \cdot 2}}$

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

$\sqrt{36 + 3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\sqrt{36 + 3^{\frac{2}{2}}}$

Divide $1$ by $1$ .

$\sqrt{36 + 3^{1}}$

Evaluate the exponent.

$\sqrt{36 + 3}$

Add $36$ and $3$ .

$\sqrt{39}$

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{\sqrt{3}}{\sqrt{39}}$ .

$\frac{\sqrt{3}}{\sqrt{39}}$

Simplify $\sin (θ)$ .

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Combine $\sqrt{3}$ and $\sqrt{39}$ into a single radical.

$\sin (θ) = \sqrt{\frac{3}{39}}$

Cancel the common factor of $3$ and $39$ .

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Factor $3$ out of $3$ .

$\sin (θ) = \sqrt{\frac{3 (1)}{39}}$

Cancel the common factors.

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Factor $3$ out of $39$ .

$\sin (θ) = \sqrt{\frac{3 \cdot 1}{3 \cdot 13}}$

Cancel the common factor.

$\sin (θ) = \sqrt{\frac{3 \cdot 1}{3 \cdot 13}}$

Rewrite the expression.

$\sin (θ) = \sqrt{\frac{1}{13}}$

Rewrite $\sqrt{\frac{1}{13}}$ as $\frac{\sqrt{1}}{\sqrt{13}}$ .

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

Any root of $1$ is $1$ .

$\sin (θ) = \frac{1}{\sqrt{13}}$

Multiply $\frac{1}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

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

Combine and simplify the denominator.

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Multiply $\frac{1}{\sqrt{13}}$ and $\frac{\sqrt{13}}{\sqrt{13}}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Divide $1$ by $1$ .

$\sin (θ) = \frac{\sqrt{13}}{13}$

Evaluate the exponent.

$\sin (θ) = \frac{\sqrt{13}}{13}$

Approximate the result.

$\sin (θ) = \frac{\sqrt{13}}{13} \approx 0.27735009$

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