40 lines
2.3 KiB
Markdown
40 lines
2.3 KiB
Markdown
# Trigonometric Identities
|
|
|
|
All of the following only apply when the denominator is not equal to zero.
|
|
|
|
$$ tan \theta = \frac{y}{x} $$
|
|
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 $1/v$.
|
|
|
|
| Base Identity | Inverse Identity | Alternate Identities | Alternate Inverse Identities |
|
|
| ----------------------------- | ------------------------------ | --------------------------------------------- | --------------------------------------------------------------------- |
|
|
| $$ sin\theta = y $$ | $$ csc\theta = \frac{1}{y} $$ | | $$ csc\theta = \frac{1}{sin\theta} $$ |
|
|
| $$ cos\theta = x $$ | $$ sec \theta = \frac{1}{x} $$ | | $$ sec\theta = \frac{1}{cos\theta} $$ |
|
|
| $$ tan\theta = \frac{y}{x} $$ | $$ cot\theta = \frac{x}{y} $$ | $$ tan\theta = \frac{sin\theta}{cos\theta} $$ | $$ cot\theta = \frac{1}{tan\theta} = \frac{cos\theta}{sin{\theta}} $$ |
|
|
|
|
$$ cot \theta = \frac{x}{y} $$
|
|
$$ sec\theta = \frac{1}{cos\theta}$$
|
|
$$ csc\theta = \frac{1}{sin\theta}$$
|
|
# 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.
|
|
$$ sin^2 \theta + cos^2 \theta = 1 $$
|
|
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**
|
|
- Sine, tangent, cosecant, and cotangent are **odd**.
|
|
## Examples
|
|
### Even and Odd Functions
|
|
> If $cot\theta = -\sqrt{3}$, what is $cot(-\theta)$?
|
|
|
|
$cot$ is an odd function, and so $cot(-\theta) = \sqrt{3}$
|
|
### Simplifying Using Identities
|
|
> Simplify $\frac{sin\theta}{cos\theta}$
|
|
|
|
1. The above equation can be split into two components
|
|
$$ \frac{sin\theta}{cos\theta} = \frac{sin\theta}{1} * \frac{1}{csc\theta} $$
|
|
2. Referring to the list of trig identities, we know that $\frac{1}{csc\theta}$ is equal to $sin\theta$.
|
|
|