Sure, let's tackle this question step-by-step.
### Objective:
We need to identify which angle completes the given half-angle formula for sine:
[tex]\[ \sin 22.5^{\circ} = -\sqrt{\frac{1 + \sin \square^{\circ}}{2}} \][/tex]
### Step-by-Step Solution:
1. Understanding the Half-Angle Formula:
The general half-angle formula for sine is:
[tex]\[
\sin \left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos \theta}{2}}
\][/tex]
2. Identifying the Formula Provided:
The given formula format is slightly different but relates to the same half-angle identity. In the form provided:
[tex]\[
\sin 22.5^{\circ} = -\sqrt{\frac{1 + \sin \square^{\circ}}{2}}
\][/tex]
Note that it deviates from the standard cosine-based formulation, but we can still deduce it using the equivalent relationships.
3. Relating the Provided Angle:
According to the standard half-angle identity, to find [tex]\(\sin 22.5^{\circ}\)[/tex]:
[tex]\[
22.5^\circ = \frac{\theta}{2} \implies \theta = 45^\circ
\][/tex]
4. Transformation and Filling the Blank:
Typically, the relationship between sine and cosine can lead us to change the trigonometric function but keep the angle calculation consistent. Here, our goal is to match the formula structure given:
For [tex]\(\theta = 45^\circ\)[/tex] we have:
[tex]\[
\sin \left(\frac{45^\circ}{2}\right) \text { or } \sin 22.5^{\circ} = -\sqrt{\frac{1 + \sin 90^\circ}{2}}
\][/tex]
This is because:
[tex]\[
\sin 90^\circ = 1
\][/tex]
Therefore, filling the blank correctly with 90 to satisfy the relationship given:
[tex]\[
\sin 22.5^{\circ} = -\sqrt{\frac{1 + \sin 90^\circ}{2}}
\][/tex]
Hence, the value that completes the square in:
[tex]\[ \sin 22.5^{\circ} = -\sqrt{\frac{1 + \sin \square^{\circ}}{2}} \][/tex]
### Answer:
[tex]\(\square = 90^\circ\)[/tex]
Equivalently, the correct angle for this half-angle identity is:
[tex]\[ \boxed{90^\circ} \][/tex]
