The question asks to find the angle for $\sin^{-1}\left(\frac{\sqrt{2}}{2}\right)$ using a calculator and to round the result to four decimal places.

Solution:

The expression $\sin^{-1}\left(\frac{\sqrt{2}}{2}\right)$ refers to the inverse sine function, also known as arcsin, which gives the angle whose sine is $\frac{\sqrt{2}}{2}$.

We know that:

$\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$

Therefore, $\sin^{-1}\left(\frac{\sqrt{2}}{2}\right) = \frac{\pi}{4}$.

In degrees, $\frac{\pi}{4}$ radians equals 45°.

Using a calculator:

To compute $\sin^{-1}\left(\frac{\sqrt{2}}{2}\right)$:

1. Set your calculator to radians or degrees, depending on your preference.
2. Input $\frac{\sqrt{2}}{2}$ into the arcsin (or inverse sine) function.

In radians: $\sin^{-1}\left(\frac{\sqrt{2}}{2}\right) = 0.7854 \text{ radians} \quad (\text{rounded to four decimal places})$

In degrees: $\sin^{-1}\left(\frac{\sqrt{2}}{2}\right) = 45^\circ$

Let me know if you need any further details or clarifications!

Here are five related questions:

1. What is the value of $\sin^{-1}(1)$?
2. How do you calculate the inverse of a cosine function?
3. What is the relationship between radians and degrees?
4. How do you compute $\cos^{-1}\left(\frac{1}{2}\right)$?
5. What are some common angles where trigonometric functions have exact values?

Tip:

To easily switch between radians and degrees on a calculator, check the settings or look for a dedicated button for this purpose.