Question

find the exact value without a calculator. double - angle formulas: $\sin(2\theta)=2\sin\theta\cos\theta$ $\cos(2\theta)=\cos^{2}\theta - \sin^{2}\theta$ $\tan(2\theta)=\frac{2\tan\theta}{1 - \tan^{2}\theta}$ half - angle formulas: $\sin(\frac{\theta}{2})=\pm\sqrt{\frac{1 - \cos\theta}{2}}$ $\cos(\frac{\theta}{2})=\pm\sqrt{\frac{1 + \cos\theta}{2}}$ $\tan(\frac{\theta}{2})=\pm\sqrt{\frac{1 - \cos\theta}{1 + \cos\theta}}$ $\sin112.5^{\circ}=\frac{\sqrt{?+\sqrt{\square}}}{\square}$

Explanation:

Step1: Identify the formula

Since \(112.5^{\circ}=\frac{225^{\circ}}{2}\), use the half - angle formula for sine: \(\sin(\frac{\theta}{2})=\pm\sqrt{\frac{1 - \cos\theta}{2}}\). Here \(\theta = 225^{\circ}\), and \(112.5^{\circ}\) is in the second quadrant where \(\sin x>0\).

Step2: Find \(\cos225^{\circ}\)

We know that \(\cos225^{\circ}=\cos(180^{\circ}+45^{\circ})=-\cos45^{\circ}=-\frac{\sqrt{2}}{2}\).

Step3: Substitute into the half - angle formula

Substitute \(\cos\theta=-\frac{\sqrt{2}}{2}\) into \(\sin(\frac{\theta}{2})=\sqrt{\frac{1 - \cos\theta}{2}}\).

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Answer:

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