Find the exact value. Write your answer using a simplified fraction and rationalize the denominator, if necessary. $$ \sin \left[\tan ^{-1}(-2)\right]=\square \quad \sqrt{\square} \quad \frac{\square}{\square} $$

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$$ \sin \left[\tan^{-1}(-2)\right] = \frac{-2\sqrt{5}}{5} \quad \sqrt{5} \quad \frac{-2}{5}. $$ To find the value of $$\sin \left[\tan^{-1}(-2)\right]$$, we start by letting $$\theta = \tan^{-1}(-2)$$. This means that $$\tan(\theta) = -2$$. We can visualize this using a right triangle. Since $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$, we can set the opposite side to be $$-2$$ and the adjacent side to be $$1$$ (the negative sign indicates that the angle is in the fourth quadrant). Now, we can find the hypotenuse $$h$$ using the Pythagorean theorem: $$ h = \sqrt{(-2)^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5}. $$ Next, we can find $$\sin(\theta)$$: $$ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{-2}{\sqrt{5}}. $$ To rationalize the denominator, we multiply the numerator and the denominator by $$\sqrt{5}$$: $$ \sin(\theta) = \frac{-2}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{-2\sqrt{5}}{5}. $$ Thus, the exact value of $$\sin \left[\tan^{-1}(-2)\right]$$ is $$ \frac{-2\sqrt{5}}{5}. $$ In the format requested, we can fill in the boxes as follows: $$ \sin \left[\tan^{-1}(-2)\right] = \frac{-2\sqrt{5}}{5} \quad \sqrt{5} \quad \frac{-2}{5}. $$ So the final answer is: $$ \sin \left[\tan^{-1}(-2)\right] = \frac{-2\sqrt{5}}{5} \quad \sqrt{5} \quad \frac{-2}{5}. $$

Find the exact value. Write your answer using a simplified fraction and rationalize the denominator, if necessary. \[ \sin \left[\tan ^{-1}(-2)\right]=\square \quad \sqrt{\square} \quad \frac{\square}{\square} \]

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Find the exact value. Write your answer using a simplified fraction and rationalize the denominator, if necessary. \[ \sin \left[\tan ^{-1}(-2)\right]=\square \quad \sqrt{\square} \quad \frac{\square}{\square} \]

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