## Triangle formulas
If two sides of a triangle are $a$ and $b$ and the angle between them is $\theta$, then the area is given by
$\text{area} = \frac{1}{2}ab\sin\theta$
## [[Trig Ratios of Special Triangles]]
### Sides
| Angles | Side 1 | Side 2 | Hypotenuse |
| -----------------------------------------------------:|:------:|:------------:|:-----------:|
| 30:60:90$\textdegree$ | $x$ | $\sqrt{3} x$ | $2x$ |
| 45:45:90$\textdegree$ | $x$ | $x$ | $\sqrt{2} x$ |
| $\frac{\pi}{6}:\frac{\pi}{3}:\frac{\pi}{2}\text{rad}$ | $x$ | $\sqrt{3} x$ | $2x$ |
| $\frac{\pi}{4}:\frac{\pi}{4}:\frac{\pi}{2}\text{rad}$ | $x$ | $x$ | $\sqrt{2}x$ |
### Resulting Special Ratios
| Function | 30$\textdegree$ | 60$\textdegree$ | 45$\textdegree$ | $\frac{\pi}{6}\text{rad}$ | $\frac{\pi}{3}\text{rad}$ | $\frac{\pi}{4}\text{rad}$ |
| -------- |:--------------------:|:--------------------:|: --------------------:|:-----------------------:|:-----------------------:|:------------------------:|
| $\sin$ | $\frac{1}{2}$ | $\frac{\sqrt{3}}{2}$ | $\frac{\sqrt{2}}{2}$ | $\frac{1}{2}$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{\sqrt{2}}$ |
| $\cos$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{2}$ | $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{2}$ | $\frac{\sqrt{2}}{2}$ |
| $\tan$ | $\frac{\sqrt{3}}{3}$ | $\sqrt{3}$ | $1$ | $\frac{\sqrt{3}}{3}$ | $\sqrt{3}$ | $1$ |
## [[Unit Circle Definitions of Trigonometric Functions#Common angles in degrees and radians|Common angles in degrees and radians]]
| angle in $\textdegree$ | angle in radians |
| :----------------------: | : ----------------: |
| 360$\textdegree$ | $2 \pi \text{ rad}$ |
| 180$\textdegree$ | $\pi \text{ rad}$ |
| 90$\textdegree$ | $\frac{\pi}{2} \text{ rad}$ |
| 60$\textdegree$ | $\frac{\pi}{3} \text{ rad}$ |
| 45$\textdegree$ | $\frac{\pi}{4} \text{ rad}$ |
| 30$\textdegree$ | $\frac{\pi}{6} \text{ rad}$ |
## Conversions
$\pi \text{ rad} = 180 \textdegree$
## Identities
### Negative Angle Identities (a.k.a “even”/“odd” identities)
$
\begin{align*}
\cos(-\theta) &= \cos \theta & \sin(-\theta) &= -\sin \theta & \tan(- \theta) &= -\tan \theta \\
\sec(-\theta) &= \sec \theta & \csc(-\theta) &= -\csc \theta & \cot(- \theta) &= -\cot \theta \\
\end{align*}
$
*Note*: $\cos$ and $\sec$ are the exception here. They are called “even” they are even functions (meaning they are symmetrical about the $y$ axis(ie $f(x)=f(-x)$). The other four are “odd” because the sign flips with the corresponding sign change of $\theta$ (ie $-f(x)=f(-x)$). Another way of thinking about this is using the [[Geometric Construction of Trigonometric Identities from the Unit Circle#ASTC Method|ASTC/CAST method]].
### [[Sum and Difference Identities]]
$
\begin{align*}
\sin(\alpha \pm \beta) &= \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \\[8pt]
\cos(\alpha \pm \beta) &= \cos \alpha \cos \beta \mp \sin \alpha \sin \beta \\[8pt]
\tan(\alpha \pm \beta) &= \frac{\tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \tan \beta}
\end{align*}
$
*Note:*
1. the order of signs( $\pm$ and $\pm$) is the same on the RHS of the sin identity
2. the order of signs( $\pm$ and $\mp$) is reversed on the RHS of the cos identity
3. In the tan identity the signs are the same in the numerator, but reversed in the denominator
### [[Double Angle Identities]]
$
\begin{align*}
\sin(2 \theta) &= 2 \sin \theta \cos \theta \\[6pt]
\cos(2 \theta) &= \cos^2 \theta - \sin^2 \theta = 2 \cos^2 \theta -1 = 1 - 2 \sin^2 \theta \\[8pt]
\tan(2 \theta) &= \frac{2 \tan \theta}{1-\tan^2 \theta}
\end{align*}
$
### [[Half Angle Identities]]
$
\begin{align*}
\sin\left( \frac{\theta}{2}\right) &= \pm \sqrt{\frac{1-\cos \theta}{2}}\\[8pt]
\cos\left( \frac{\theta}{2}\right) &= \pm \sqrt{\frac{1+\cos \theta}{2}}\\[8pt]
\tan\left( \frac{\theta}{2}\right) &= \pm \sqrt{\frac{1-\cos \theta}{1+\cos \theta}} \\[8pt]
&= \frac{1-\cos \theta}{\sin\theta} \\[8pt]
&= \frac{\sin \theta}{1 + \cos \theta} \\[8pt]
\end{align*}
$
*note:* The second two forms of $\tan(\theta/2)$ can be derived by multiplying the top and bottom of the first form by $1-\cos \theta$ and $1+\cos \theta$ respectively
### [[Pythagorean Trigonometric Identities|Pythagorean Identities]]
$
\begin{align*}
\sin^2 \theta + \cos^2 \theta &= 1\\
\csc^2 \theta - \cot^2 \theta &= 1 \\
\sec^2 \theta - \tan^2 \theta &= 1 \\
\end{align*}
$
### [[Trigonometric Functions of Complementary Angles|Complementary Angles]]
$
\begin{align*}
\sin \theta &= \cos(\frac{\pi}{2} - \theta) \\
\cos \theta &= \sin(\frac{\pi}{2} - \theta) \\
\\
\sec \theta &= \csc(\frac{\pi}{2} - \theta) \\
\csc \theta &= \sec(\frac{\pi}{2} - \theta) \\
\\
\tan \theta &= \cot(\frac{\pi}{2} - \theta) \\
\cot \theta &= \tan(\frac{\pi}{2} - \theta) \\
\end{align*}
$
### [[Geometric Construction of Trigonometric Identities from the Unit Circle|Unit Circle Identities]]
$
\begin{align*}
\cos(\pi - \theta) &= -\cos \theta & \sin(\pi - \theta) &= \sin \theta \\
\cos(\pi + \theta) &= -\cos \theta & \sin(\pi + \theta) &= -\sin \theta \\
\tan(\pi + \theta) &= \tan \theta & \tan(\pi - \theta) &= -\tan \theta
\end{align*}
$
### Power-reducing formulas
*Note*: These are just another way of stating the [[#Half Angle Identities]].
$
\begin{align*}
\sin^2 \theta &= \frac{1-\cos(2\theta)}{2} \\[8pt]
\cos^2 \theta &= \frac{1+\cos(2\theta)}{2} \\[8pt]
\tan^2 \theta &= \frac{1-\cos(2\theta)}{1+\cos(2\theta)} \\[8pt]
\end{align*}
$
### Product sum formulas
$
\begin{align*}
\sin \alpha \sin \beta &= \frac{\cos(\alpha-\beta) - \cos(\alpha + \beta)}{2} \\[8pt]
\cos \alpha \cos \beta &= \frac{\cos(\alpha-\beta) + \cos(\alpha + \beta)}{2} \\[8pt]
\sin \alpha \cos \beta &= \frac{\sin(\alpha+\beta) - \sin(\alpha - \beta)}{2} \\[8pt]
\cos \alpha \sin \beta &= \frac{\sin(\alpha+\beta) + \sin(\alpha - \beta)}{2} \\[8pt]
\end{align*}
$
## Laws
### Law of sines
$
\frac{\sin \alpha}{a} = \frac{\sin \beta}{b} = \frac{\sin \gamma}{c}
$
### Law of cosines
$
c^2 = a^2 + b^2 -2ab \cos \theta
$
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