Find the Sine (-2,- square root of 2)

$(- 2, - \sqrt{2})$

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

Opposite : $- \sqrt{2}$

Adjacent : $- 2$

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

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Simplify the expression.

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

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

Apply the product rule to $- \sqrt{2}$ .

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

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

$\sqrt{4 + 1 {\sqrt{2}}^{2}}$

Multiply ${\sqrt{2}}^{2}$ by $1$ .

$\sqrt{4 + {\sqrt{2}}^{2}}$

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

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

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

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

$\sqrt{4 + 2^{\frac{1}{2} \cdot 2}}$

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

$\sqrt{4 + 2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\sqrt{4 + 2^{\frac{2}{2}}}$

Divide $1$ by $1$ .

$\sqrt{4 + 2^{1}}$

Evaluate the exponent.

$\sqrt{4 + 2}$

Add $4$ and $2$ .

$\sqrt{6}$

$\sin (θ) = \frac{Opposite}{Hypotenuse}$ therefore $\sin (θ) = \frac{- \sqrt{2}}{\sqrt{6}}$ .

$\frac{- \sqrt{2}}{\sqrt{6}}$

Simplify $\sin (θ)$ .

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Combine $\sqrt{2}$ and $\sqrt{6}$ into a single radical.

$\sin (θ) = - \sqrt{\frac{2}{6}}$

Cancel the common factor of $2$ and $6$ .

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Factor $2$ out of $2$ .

$\sin (θ) = - \sqrt{\frac{2 (1)}{6}}$

Cancel the common factors.

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Factor $2$ out of $6$ .

$\sin (θ) = - \sqrt{\frac{2 \cdot 1}{2 \cdot 3}}$

Cancel the common factor.

$\sin (θ) = - \sqrt{\frac{2 \cdot 1}{2 \cdot 3}}$

Rewrite the expression.

$\sin (θ) = - \sqrt{\frac{1}{3}}$

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

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

Any root of $1$ is $1$ .

$\sin (θ) = - \frac{1}{\sqrt{3}}$

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Combine and simplify the denominator.

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Multiply $\frac{1}{\sqrt{3}}$ and $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\sin (θ) = - \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Raise $\sqrt{3}$ to the power of $1$ .

$\sin (θ) = - \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Raise $\sqrt{3}$ to the power of $1$ .

$\sin (θ) = - \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

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

$\sin (θ) = - \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

$\sin (θ) = - \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

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

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

$\sin (θ) = - \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

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

$\sin (θ) = - \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

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

$\sin (θ) = - \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\sin (θ) = - \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Divide $1$ by $1$ .

$\sin (θ) = - \frac{\sqrt{3}}{3}$

Evaluate the exponent.

$\sin (θ) = - \frac{\sqrt{3}}{3}$

Approximate the result.

$\sin (θ) = - \frac{\sqrt{3}}{3} \approx - 0.57735026$

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