Certainly! We are tasked with calculating the value of [tex]\(\sin 22.5^\circ\)[/tex].
To do this, we can use a trigonometric identity for the sine of a half-angle. Specifically, we will use the identity:
[tex]\[ \sin \left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos \theta}{2}} \][/tex]
Let's start by recognizing that [tex]\(22.5^\circ\)[/tex] is half of [tex]\(45^\circ\)[/tex], so we can set [tex]\(\theta = 45^\circ\)[/tex].
Now, we apply the half-angle identity for sine:
[tex]\[ \sin 22.5^\circ = \sin \left(\frac{45^\circ}{2}\right) = \sqrt{\frac{1 - \cos 45^\circ}{2}} \][/tex]
Next, we need the value of [tex]\(\cos 45^\circ\)[/tex]. We know from basic trigonometric values that:
[tex]\[ \cos 45^\circ = \frac{\sqrt{2}}{2} \][/tex]
Now, substituting [tex]\(\cos 45^\circ\)[/tex] into the half-angle identity:
[tex]\[ \sin 22.5^\circ = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} \][/tex]
Simplify the expression inside the square root:
[tex]\[ \sin 22.5^\circ = \sqrt{\frac{2 - \sqrt{2}}{4}} \][/tex]
[tex]\[ \sin 22.5^\circ = \sqrt{\frac{2 - \sqrt{2}}{2} \cdot \frac{1}{2}} \][/tex]
[tex]\[ \sin 22.5^\circ = \sqrt{\frac{2 - \sqrt{2}}{2}} \cdot \frac{1}{\sqrt{2}} \][/tex]
Finally, simplifying further:
[tex]\[ \sin 22.5^\circ = \sqrt{\frac{1 - \cos 45^\circ}{2}} \][/tex]
Hence, the numerical value of [tex]\(\sin 22.5^\circ\)[/tex] comes out to be:
[tex]\[ \sin 22.5^\circ \approx 0.3826834323650897 \][/tex]
Thus:
[tex]\[ \sin 22.5^\circ \approx 0.3826834323650897 \][/tex]
This is the final value.
