Questions: Find the exact value of the following expression. sin^(-1)(sqrt(2)/2) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. sin^(-1)(sqrt(2)/2) = □ (Simplify your answer. Type an exact answer, using π as needed. Use integers or fractions for any numbers in the expression) B. The function is not defined.

Solution Steps

To find the exact value of \(\sin^{-1}\left(\frac{\sqrt{2}}{2}\right)\), we need to determine the angle whose sine is \(\frac{\sqrt{2}}{2}\). This is a common value in trigonometry, and we know that \(\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\). Therefore, \(\sin^{-1}\left(\frac{\sqrt{2}}{2}\right) = \frac{\pi}{4}\).

We need to find the angle \(\theta\) such that \(\sin(\theta) = \frac{\sqrt{2}}{2}\). This is a common trigonometric value.

From trigonometric identities, we know that: \[ \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \] Thus, \(\sin^{-1}\left(\frac{\sqrt{2}}{2}\right) = \frac{\pi}{4}\).

Using the Python output: \[ \text{value} = 0.7854 \quad (\text{rounded to four significant digits}) \] \[ \text{result} = 0.2500 \quad (\text{rounded to four significant digits}) \] This confirms that: \[ \sin^{-1}\left(\frac{\sqrt{2}}{2}\right) = 0.25 \times \pi = \frac{\pi}{4} \]

Final Answer