24 lines
1.3 KiB
Markdown
24 lines
1.3 KiB
Markdown
Given the formula $\sin(\alpha + \beta)$:
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$$ \sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta) $$
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$$ \sin(\alpha - \beta) = \sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta) $$
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Given the formula $\cos(\alpha + \beta)$:
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$$ \cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta) $$
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$$ \cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta) $$
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Given the formula $\tan(\alpha + \beta)$:
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$$\tan(\alpha + \beta) = \dfrac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta} $$
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$$\tan(\alpha - \beta) = \dfrac{\tan\alpha - \tan\beta}{1 + \tan\alpha\tan\beta} $$
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## Cofunctions
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Given that cofunctions are two functions that add up to 90 degrees, you can use the trig identities for sum and difference to find cofunctions.
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For a right triangle where $\alpha = \theta$, $\beta = \frac{\pi}{2} - \theta$.
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This means that $\sin(\theta) = \cos(\frac{\pi}{2} - \theta)$
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Using this information, you can derive various cofunction identities.
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| $\sin\theta = \cos(\frac{\pi}{2} - \theta)$ | $\cos\theta = \sin(\frac{\pi}{2} - \theta)$ |
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| ------------------------------------------- | -------------------------------------------- |
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| $\tan\theta = \cot(\frac{\pi}{2} - \theta)$ | $\cot\theta = \tan(\frac{\pi}{2} - \theta))$ |
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| $\sec\theta = \csc(\frac{\pi}{2} - \theta)$ | $\csc\theta = \sec(\frac{\pi}{2} - \theta)$ |
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