2024-10-16 16:07:57 +00:00
|
|
|
Given the formula $\sin(\alpha + \beta)$:
|
2024-10-16 16:18:30 +00:00
|
|
|
$$ \sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta) $$
|
2024-10-23 02:46:27 +00:00
|
|
|
$$ \sin(\alpha - \beta) = \sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta) $$
|
2024-10-16 16:18:30 +00:00
|
|
|
Given the formula $\cos(\alpha + \beta)$:
|
2024-10-16 17:23:24 +00:00
|
|
|
$$ \cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta) $$
|
2024-10-23 02:46:27 +00:00
|
|
|
$$ \cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta) $$
|
2024-10-23 02:41:27 +00:00
|
|
|
Given the formula $\tan(\alpha + \beta)$:
|
2024-10-23 02:46:27 +00:00
|
|
|
$$\tan(\alpha + \beta) = \dfrac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta} $$
|
|
|
|
|
$$\tan(\alpha - \beta) = \dfrac{\tan\alpha - \tan\beta}{1 + \tan\alpha\tan\beta} $$
|
2024-10-24 16:23:18 +00:00
|
|
|
|
|
|
|
|
## Cofunctions
|
|
|
|
|
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.
|
|
|
|
|
|
|
|
|
|
For a right triangle where $\alpha = \theta$, $\beta = \frac{\pi}{2} - \theta$.
|
|
|
|
|
|
|
|
|
|
This means that $\sin(\theta) = \cos(\frac{\pi}{2} - \theta)$
|
|
|
|
|
|
|
|
|
|
Using this information, you can derive various cofunction identities.
|
|
|
|
|
|
|
|
|
|
| $\sin\theta = \cos(\frac{\pi}{2} - \theta)$ | $\cos\theta = \sin(\frac{\pi}{2} - \theta)$ |
|
|
|
|
|
| ------------------------------------------- | -------------------------------------------- |
|
|
|
|
|
| $\tan\theta = \cot(\frac{\pi}{2} - \theta)$ | $\cot\theta = \tan(\frac{\pi}{2} - \theta))$ |
|
|
|
|
|
| $\sec\theta = \csc(\frac{\pi}{2} - \theta)$ | $\csc\theta = \sec(\frac{\pi}{2} - \theta)$ |
|
