Questions: Find the exact value (in radians). arccsc(√2)=

Transcript text: Find the exact value (in radians). $\operatorname{arccsc}(\sqrt{2})=$ $\square$

Solution

Solution Steps

To find the exact value of $\operatorname{arccsc}(\sqrt{2})$, we need to determine the angle $\theta$ such that $\csc(\theta) = \sqrt{2}$. This implies that $\sin(\theta) = \frac{1}{\sqrt{2}}$. We know that $\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$, which is equivalent to $\frac{1}{\sqrt{2}}$. Therefore, $\theta = \frac{\pi}{4}$ is the angle we are looking for, considering the range of $\operatorname{arccsc}$.

Step 1: Understanding the Problem

We need to find the exact value of $\operatorname{arccsc}(\sqrt{2})$. This requires us to determine the angle $\theta$ such that $\csc(\theta) = \sqrt{2}$.

Step 2: Relating Cosecant to Sine

The cosecant function is the reciprocal of the sine function, so we have: \[ \csc(\theta) = \sqrt{2} \implies \sin(\theta) = \frac{1}{\sqrt{2}} \]

Step 3: Finding the Angle

We know that: \[ \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} = \frac{1}{\sqrt{2}} \] Thus, the angle $\theta$ that satisfies this equation is: \[ \theta = \frac{\pi}{4} \]

Final Answer

The exact value of $\operatorname{arccsc}(\sqrt{2})$ is $\boxed{\frac{\pi}{4}}$.

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