Question

use the half - angle formulas to find the exact value of the trigonometric function \\( \sin ( - 22.5 ^ { \circ } ) \\).
\\( \sin ( - 22.5 ^ { \circ } ) = \square \\)
(type an exact answer, using radicals as needed.)

Explanation:

Step1: Use the property of sine function

Since \(\sin(-\alpha)=-\sin\alpha\), then \(\sin(- 22.5^{\circ})=-\sin(22.5^{\circ})\). And \(22.5^{\circ}=\frac{45^{\circ}}{2}\).
The half - angle formula for sine is \(\sin\frac{\alpha}{2}=\pm\sqrt{\frac{1 - \cos\alpha}{2}}\). Here \(\alpha = 45^{\circ}\), and \(22.5^{\circ}\) is in the first quadrant, but we have \(\sin(-22.5^{\circ})\), so we use the negative sign.

Step2: Substitute \(\alpha = 45^{\circ}\) into the half - angle formula

We know that \(\cos45^{\circ}=\frac{\sqrt{2}}{2}\). Substitute into \(\sin(-22.5^{\circ})=-\sin(22.5^{\circ})=-\sqrt{\frac{1-\cos45^{\circ}}{2}}\)

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Answer:

\(-\frac{\sqrt{2 - \sqrt{2}}}{2}\)