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The double-angle formulas are very useful for simplifying trigonometric expressions where angles are doubled. For sine, the formula is: \(\text{sin}(2 \theta) = 2 \text{sin}(\theta) \text{cos}(\theta)\). These formulas can help you find values for trigonometric functions without needing a calculator.
In our example, we used \(\text{sin}(2 \theta)\) when \( \theta = \frac{\pi}{8} \). By substituting \( \frac{\pi}{8} \) into the double-angle formula, we get \( \text{sin}(2 \theta) = 2 \text{sin}( \frac{\pi}{8}) \text{cos}(\frac{\pi}{8}) \).
This leads us to use the identities of \(\text{sin} \) and \(\text{cos} \) for further evaluation.

The half-angle formulas are another powerful set of trigonometric identities. They are derived from the double-angle formulas and can be used to find the sine and cosine values of half-angles. The formulas are:
\( \text{sin} \frac{x}{2} = \pm \sqrt{\frac{1 - \text{cos}(x)}{2}} \)
and
\( \text{cos} \frac{x}{2} = \pm \sqrt{\frac{1 + \text{cos}(x)}{2}} \).
In our example, we need \( \text{sin}( \frac{\pi}{8} ) \) and \( \text{cos}( \frac{\pi}{8} ) \). Because \( \frac{\pi}{8} = \frac{\pi}{4}/2 \), we used half-angle formulas:
\( \text{sin} \frac{\pi}{8} = \sqrt{\frac{1 - \text{cos}(\frac{\pi}{4})}{2}} \) and
\( \text{cos} \frac{\pi}{8} = \sqrt{\frac{1 + \text{cos}(\frac{\pi}{4})}{2}} \).
This helps to find the necessary sine and cosine values to be used in the double-angle formula.

Finding the exact values for sine and cosine functions involves understanding key angles and their corresponding values. Here, knowing that \( \text{cos}(\frac{\pi}{4}) = \frac{\text{\backsqrt{2}}}{2} \) is crucial.
Using the half-angle formulas:
\( \text{sin} \frac{\pi}{8} = \sqrt{\frac{1 - \frac{\text{\backsqrt{2}}}{2}}{2}} = \frac{\text{\backsqrt{2 - \text{\backsqrt{2}}}}}{2} \) and
\( \text{cos} \frac{\pi}{8} = \sqrt{\frac{1 + \frac{\text{\backsqrt{2}}}{2}}{2}} = \frac{\text{\backsqrt{2 + \text{\backsqrt{2}}}}}{2} \).
Finally, plug these into the double-angle formula to simplify and get:
\( \text{sin}(2 \theta) = \frac{\text{\backsqrt{2}}}{2} \).
Understanding these steps helps you grasp trigonometric identities and apply them to new problems.