9.2.1: Resources and Key Concepts — Read

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Resources

Sum and Difference Identities for the Cosine: For any angles \(\alpha\) and \(\beta\):
- \(\cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)\)
- \(\cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta)\)
Sum and Difference Identities for the Sine: For any angles \(\alpha\) and \(\beta\):
- \(\sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)\)
- \(\sin(\alpha - \beta) = \sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta)\)
Sum and Difference Identities for the Tangent: For any angles \(\alpha\) and \(\beta\) where the functions are defined:
- \(\tan(\alpha + \beta) = \frac{\tan(\alpha) + \tan(\beta)}{1 - \tan(\alpha)\tan(\beta)}\)
- \(\tan(\alpha - \beta) = \frac{\tan(\alpha) - \tan(\beta)}{1 + \tan(\alpha)\tan(\beta)}\)

You Cannot "Split" a Trigonometric Function Across its Arguments: A very common mistake is to assume that a trigonometric function distributes over a sum or difference. For example, \(\cos(\alpha + \beta)\) is not equal to \(\cos(\alpha) + \cos(\beta)\). The Sum and Difference Identities must be used instead.
Confusing the Signs in the Identities: The signs in the Sum and Difference Identities can be tricky. For the cosine identities, the sign in the middle of the expression is the opposite of the operation in the argument (e.g., \(\cos(\alpha + \beta)\) has a minus sign). For the sine identities, the sign in the middle matches the operation in the argument (e.g., \(\sin(\alpha + \beta)\) has a plus sign).

This page titled 9.2.1: Resources and Key Concepts was last modified on Tue, 08 Jul 2025 17:45:39 GMT and is shared under a not declared license and was authored, remixed, and/or curated by Roy Simpson.