use a half - angle formula to find the exact value of the following expression. sin 67.5° sin 67.5° = □…

Explanation:

Step1: Identify the half - angle formula

The half - angle formula for sine is $\sin\frac{\alpha}{2}=\pm\sqrt{\frac{1 - \cos\alpha}{2}}$. Since $67.5^{\circ}=\frac{135^{\circ}}{2}$ and $67.5^{\circ}$ is in the first quadrant (where sine is positive), we use the positive form of the formula. Here $\alpha = 135^{\circ}$.

Step2: Find the value of $\cos\alpha$

We know that $\cos135^{\circ}=-\frac{\sqrt{2}}{2}$.

Step3: Substitute into the half - angle formula

Substitute $\cos\alpha=-\frac{\sqrt{2}}{2}$ into $\sin\frac{\alpha}{2}=\sqrt{\frac{1 - \cos\alpha}{2}}$. [ \begin{align*} \sin67.5^{\circ}&=\sqrt{\frac{1-(-\frac{\sqrt{2}}{2})}{2}}\ &=\sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}\ &=\sqrt{\frac{\frac{2+\sqrt{2}}{2}}{2}}\ &=\sqrt{\frac{2 + \sqrt{2}}{4}}\ &=\frac{\sqrt{2+\sqrt{2}}}{2} \end{align*} ]

Answer:

$\frac{\sqrt{2+\sqrt{2}}}{2}$