use the half - angle formulas to find the exact value of the trigonometric function \\( \\sin ( - 22.5 ^ {…

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

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

Answer

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}}) [ \begin{align*} \sin(-22.5^{\circ})&=-\sqrt{\frac{1-\frac{\sqrt{2}}{2}}{2}}\ &=-\sqrt{\frac{2 - \sqrt{2}}{4}}\ &=-\frac{\sqrt{2-\sqrt{2}}}{2} \end{align*} ]

Answer:

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