Questions: Find the exact value of the expression. sin [tan^(-1)(1)] Select the correct choice and fill in any answer boxes in your choice below. A. sin [tan^(-1)(1)]= (Simplify your answer, including any radicals. Use integers or fractions B. There is no solution.

Solution Steps

To find the exact value of \(\sin \left[\tan^{-1}(1)\right]\), we need to understand the relationship between the trigonometric functions involved. The expression \(\tan^{-1}(1)\) represents an angle whose tangent is 1. We need to determine this angle and then find the sine of that angle.

1. Determine the angle \(\theta\) such that \(\tan(\theta) = 1\).
2. Recognize that \(\theta = \frac{\pi}{4}\) because \(\tan(\frac{\pi}{4}) = 1\).
3. Calculate \(\sin(\frac{\pi}{4})\).

We need to find the angle \(\theta\) such that \(\tan(\theta) = 1\). The angle that satisfies this condition is: \[ \theta = \tan^{-1}(1) = \frac{\pi}{4} \]

Next, we calculate the sine of the angle \(\theta\): \[ \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \]

Final Answer