To find the exact value of cos 225 degrees using the half angle formula, we can start by recognizing that 225 degrees is actually half of 450 degrees. The half angle formula for cosine is given by:
\[
ext{cos} \left( \frac{\theta}{2} \right) = \pm \sqrt{\frac{1 + \text{cos} \theta}{2}}
\]
In our case, we will use
ci 450 degrees. First, we need to find the cosine of 450 degrees. Since 450 degrees is more than 360 degrees, we can reduce it:
\[ 450 – 360 = 90 \text{ degrees} \]
The cosine of 90 degrees is 0. Now we can apply the half angle formula. Using \( \theta = 450 \) degrees, we substitute it into the formula:
\[
\text{cos} 225 = \text{cos} \left( \frac{450}{2} \right) = \sqrt{\frac{1 + \text{cos} 450}{2}}
\]
Substituting \( \text{cos} 450 = 0 \) into the equation gives us:
\[
\text{cos} 225 = \sqrt{\frac{1 + 0}{2}} = \sqrt{\frac{1}{2}}
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\]
This simplifies to:
\[ ext{cos} 225 = \frac{\sqrt{2}}{2}
However, since 225 degrees is in the third quadrant where cosine values are negative, we need to take the negative value:
\[ ext{cos} 225 = -\frac{\sqrt{2}}{2} \]
So the exact value of cos 225 degrees, using the half angle formula, is:
\[ ext{cos} 225 = -\frac{\sqrt{2}}{2} \]