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education/math/MATH1060 (trig)/Identities.md
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@ -5,15 +5,23 @@ All of the following only apply when the denominator is not equal to zero.
$$ tan \theta = \frac{y}{x} $$ $$ tan \theta = \frac{y}{x} $$
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 $1/v$. 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 $1/v$.
| Base Identity | Inverse Identity | Alternate Identities | Alternate Inverse Identities | | Base Identity | Inverse Identity | Alternate Identities | Alternate Inverse Identities |
| ----------------------------- | ------------------------------ | --------------------------------------------- | ------------------------------------------------------------------------- | | ----------------------------- | ------------------------------ | --------------------------------------------- | --------------------------------------------------------------------- |
| $$ sin\theta = y $$ | $$ csc\theta = \frac{1}{y} $$ | | $$ csc\theta = \frac{1}{sin\theta} $$ | | $$ sin\theta = y $$ | $$ csc\theta = \frac{1}{y} $$ | | $$ csc\theta = \frac{1}{sin\theta} $$ |
| $$ cos\theta = x $$ | $$ sec \theta = \frac{1}{x} $$ | | | | $$ cos\theta = x $$ | $$ sec \theta = \frac{1}{x} $$ | | $$ sec\theta = \frac{1}{cos\theta} $$ |
| $$ tan\theta = \frac{y}{x} $$ | $$ cot\theta = \frac{x}{y} $$ | $$ tan\theta = \frac{sin\theta}{cos\theta} $$ | <br>$$ cot\theta = \frac{1}{tan\theta} = \frac{cos\theta}{sin{\theta}} $$ | | $$ tan\theta = \frac{y}{x} $$ | $$ cot\theta = \frac{x}{y} $$ | $$ tan\theta = \frac{sin\theta}{cos\theta} $$ | $$ cot\theta = \frac{1}{tan\theta} = \frac{cos\theta}{sin{\theta}} $$ |
| | | | |
$$ cot \theta = \frac{x}{y} $$ $$ cot \theta = \frac{x}{y} $$
$$ sec\theta = \frac{1}{cos\theta}$$ $$ sec\theta = \frac{1}{cos\theta}$$
$$ csc\theta = \frac{1}{sin\theta}$$ $$ csc\theta = \frac{1}{sin\theta}$$
# Pythagorean Identities # Pythagorean Identities
The Pythagorean identity expresses the Pythagorean theorem in terms of trigonometric functions. It's a basic relation between the sine and cosine functions. The Pythagorean identity expresses the Pythagorean theorem in terms of trigonometric functions. It's a basic relation between the sine and cosine functions.
$$ sin^2 \theta + cos^2 \theta = 1 $$ $$ sin^2 \theta + cos^2 \theta = 1 $$
There are more forms that are useful, but they can be derived from the above formula:
$$ 1 + tan^2\theta = sec^2\theta $$
$$ cot^2 \theta + 1 = csc^2\theta $$
# Even and Odd Identities
- A function is even if $f(-x) = f(x)$.
- A function is odd if $f(-x) = -f(x)$
Cosine and secant are **even*
## Examples