Turnbull Rodriquez
11/15/2023 · High School
Find the exact value. \[ 2 \arcsin \frac{\sqrt{2}}{2}-\frac{\pi}{6}=\square \quad \pi \quad \frac{\square}{\square} \sqrt{\square} \]
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Tutor-Verified Answer
Step-by-step Solution
To solve the expression
\[
2 \arcsin \frac{\sqrt{2}}{2} - \frac{\pi}{6},
\]
we first need to evaluate \( \arcsin \frac{\sqrt{2}}{2} \).
The value of \( \arcsin \frac{\sqrt{2}}{2} \) corresponds to the angle whose sine is \( \frac{\sqrt{2}}{2} \). This angle is
\[
\frac{\pi}{4}.
\]
Now we can substitute this value back into the expression:
\[
2 \arcsin \frac{\sqrt{2}}{2} = 2 \cdot \frac{\pi}{4} = \frac{\pi}{2}.
\]
Next, we substitute this result into the original expression:
\[
2 \arcsin \frac{\sqrt{2}}{2} - \frac{\pi}{6} = \frac{\pi}{2} - \frac{\pi}{6}.
\]
To perform this subtraction, we need a common denominator. The least common multiple of 2 and 6 is 6. We can rewrite \( \frac{\pi}{2} \) as follows:
\[
\frac{\pi}{2} = \frac{3\pi}{6}.
\]
Now we can perform the subtraction:
\[
\frac{3\pi}{6} - \frac{\pi}{6} = \frac{3\pi - \pi}{6} = \frac{2\pi}{6} = \frac{\pi}{3}.
\]
Thus, the exact value of the expression is
\[
\frac{\pi}{3}.
\]
In the format given in the problem, we can express this as:
\[
\frac{\square}{\square} \sqrt{\square} = \frac{1}{3} \sqrt{1}.
\]
Therefore, the final answer is
\[
\frac{\pi}{3}.
\]
Quick Answer
\(\frac{\pi}{3}\)
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