33 lines
2.0 KiB
Markdown
33 lines
2.0 KiB
Markdown
# Trigonometric Identities
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All of the following only apply when the denominator is not equal to zero.
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$$ tan \theta = \frac{y}{x} $$
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Because the following are inverses of their counterparts, you only need to remember the equivalents for $sin$, $cos$, and $tan$, then just find the inverse by taking $1/v$.
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| Base Identity | Inverse Identity | Alternate Identities | Alternate Inverse Identities |
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| ----------------------------- | ------------------------------ | --------------------------------------------- | --------------------------------------------------------------------- |
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| $$ sin\theta = y $$ | $$ csc\theta = \frac{1}{y} $$ | | $$ csc\theta = \frac{1}{sin\theta} $$ |
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| $$ cos\theta = x $$ | $$ sec \theta = \frac{1}{x} $$ | | $$ sec\theta = \frac{1}{cos\theta} $$ |
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| $$ tan\theta = \frac{y}{x} $$ | $$ cot\theta = \frac{x}{y} $$ | $$ tan\theta = \frac{sin\theta}{cos\theta} $$ | $$ cot\theta = \frac{1}{tan\theta} = \frac{cos\theta}{sin{\theta}} $$ |
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$$ cot \theta = \frac{x}{y} $$
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$$ sec\theta = \frac{1}{cos\theta}$$
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$$ csc\theta = \frac{1}{sin\theta}$$
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# Pythagorean Identities
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The Pythagorean identity expresses the Pythagorean theorem in terms of trigonometric functions. It's a basic relation between the sine and cosine functions.
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$$ sin^2 \theta + cos^2 \theta = 1 $$
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There are more forms that are useful, but they can be derived from the above formula:
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$$ 1 + tan^2\theta = sec^2\theta $$
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$$ cot^2 \theta + 1 = csc^2\theta $$
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# Even and Odd Identities
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- A function is even if $f(-x) = f(x)$.
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- A function is odd if $f(-x) = -f(x)$
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- Cosine and secant are **even**
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- Sine, tangent, cosecant, and cotangent are **odd**.
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## Examples
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> If $cot\theta = -\sqrt{3}$, what is $cot(-\theta)$?
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$cot$ is an odd function, and so $cot(-\theta) = \sqrt{3}$
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