6.4.1: Resources and Key Concepts
- Page ID
- 196939
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Keywords
- Power Reduction Identities
- Half-Angle Identities
Key Concepts
Theorems
- Power Reduction Identities:
- \(\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}\)
- \(\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}\)
- \(\tan^2(\theta) = \frac{1 - \cos(2\theta)}{1 + \cos(2\theta)}\)
- Half-Angle Identities:
- \(\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos(\theta)}{2}}\)
- \(\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos(\theta)}{2}}\)
- \(\tan\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}} = \frac{\sin(\theta)}{1 + \cos(\theta)} = \frac{1 - \cos(\theta)}{\sin(\theta)}\)
Common Mistakes
- Forgetting the \(\pm\) Sign: The Half-Angle Identities for sine and cosine include a \(\pm\) symbol. This does not mean both signs are correct. You must determine the correct sign (positive or negative) based on the quadrant in which the half-angle \(\frac{\theta}{2}\) terminates.
- Determining the Sign from the Wrong Angle: The sign for a half-angle identity, like \(\sin(\frac{\theta}{2})\), depends on the quadrant of \(\frac{\theta}{2}\), not the quadrant of the original angle \(\theta\).
- Incorrectly Applying Power Reduction: The Power Reduction Identities are used to rewrite even powers of sine and cosine in terms of the first power of cosine. They cannot be used directly on odd powers.