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Trigonometry: (lesson 1 of 3)

## Trigonometric Formulas

\begin{aligned} \sin \alpha &= \frac{opposite}{hypotenuse} \\ \cos \alpha &= \frac{adjacent}{hypotenuse} \\ \tan \alpha &= \frac{opposite}{adjacent} \\ \cot \alpha &= \frac{adjacent}{opposite} \\ \sec \alpha &= \frac{hypotenuse}{adjacent} \\ \csc \alpha &= \frac{hypotenuse}{opposite} \end{aligned}

### Reciprocal Properties:

\begin{aligned} \cot (x) &= \frac{1}{\tan (x)} \\ \csc (x) &= \frac{1}{\sin (x)} \\ \sec (x) &= \frac{1}{\cos (x)} \\ \end{aligned} \begin{aligned} \tan (x) \cot (x) &= 1 \\ \sin (x) \csc (x) &= 1 \\ \cos (x) \sec (x) &= 1 \end{aligned}

### Quotient Properties:

\begin{aligned} \tan (x) &= \frac{\sin (x)}{\cos (x)} \\ \cot (x) &= \frac{\cos (x)}{\sin (x)} \\ \tan (x) &= \frac{\sec (x)}{\csc (x)} \\ \cot (x) &= \frac{\csc (x)}{\sec (x)} \\ \tan (x) &= \frac{\sec (x)}{\csc (x)} \\ \cot (x) &= \frac{\csc (x)}{\sec (x)} \end{aligned}

### Odd/Even Identities

\begin{aligned} \sin (-x) &= - \sin (x) \\ \cos (-x) &= - \cos (x) \\ \tan (-x) &= - \tan (x) \\ \csc (-x) &= - \csc (x) \\ \sec (-x) &= - \sec (x) \\ \cot (-x) &= - \cot (x) \end{aligned}

\begin{aligned} \sin ( \frac{\pi }{2} - x ) &= \cos (x) \\ \cos ( \frac{\pi }{2} - x ) &= \sin (x) \\ \tan ( \frac{\pi }{2} - x ) &= \cot (x) \\ \cot ( \frac{\pi }{2} - x ) &= \tan (x) \end{aligned}

### Cofunction Identities - degrees

\begin{aligned} \sin ( 90^\circ - x ) &= \cos (x) \\ \cos ( 90^\circ - x ) &= \sin (x) \\ \tan ( 90^\circ - x ) &= \cot (x) \\ \cot ( 90^\circ - x ) &= \tan (x) \end{aligned}

\begin{aligned} \sin ( x + 2 \pi ) &= \sin (x) \\ \cos ( x + 2 \pi ) &= \cos (x) \\ \tan ( x + \pi ) &= \tan (x) \\ \cot ( x + \pi ) &= \cot (x) \end{aligned}

### Periodicity Identities - degrees

\begin{aligned} \sin ( x + 360^\circ ) &= \sin (x) \\ \sin ( x + 360^\circ ) &= \cos (x) \\ \tan ( x + 180^\circ ) &= \tan (x) \\ \cot ( x + 180^\circ ) &= \cot (x) \end{aligned}

### Sum/Difference Identities

\begin{aligned} \sin (x + y) &= \sin (x) \cos (y) + \cos (x) \sin (y) \\ \cos (x + y) &= \cos (x) \cos (y) - \sin (x) \sin (y) \\ \tan (x + y) &= \frac{\tan (x) + \tan y}{1 - \tan (x) \cdot \tan (y)} \\ \sin (x - y) &= \sin (x) \cos (y) - \cos (x) \sin (y) \\ \cos (x - y) &= \cos (x) \cos (y) + \sin (x) \sin (y) \\ \tan (x - y) &= \frac{\tan (x) - \tan (y)}{1 + \tan (x) \cdot \tan (y)} \end{aligned}

### Double Angle Identities

\begin{aligned} \sin(2x) &= 2 \sin (x) \cos (x) \\ \cos(2x) &= \cos^2 (x) - \sin^2(x) \\ \cos(2x) &= 2 \cos^2(x) - 1 \\ \cos(2x) &= 1 - 2 \sin^2(x) \\ \tan(2x) &= [2 \tan(x)] / [1 - \tan^2(x)] \end{aligned}

### Half Angle Identities

\begin{aligned} \sin ( \frac{x}{2} ) &= \pm \sqrt {\frac{1 - \cos (x)}{2}} \\ \cos ( \frac{x}{2} ) &= \pm \sqrt {\frac{1 + \cos (x)}{2}} \\ \cos ( \frac{x}{2} ) &= \pm \sqrt {\frac{1 - \cos (x)}{1 + \cos (x)}} \end{aligned}

### Product identities

\begin{aligned} \sin (x) \cdot \cos (y) &= \frac{\sin (x + y) + \sin (x - y)}{2} \\ \cos (x) \cdot \cos (y) &= \frac{\cos (x + y) + \cos (x - y)}{2} \\ \sin (x) \cdot \sin (y) &= \frac{\cos (x + y) - \cos (x - y)}{2} \end{aligned}

### Sum to Product Identities

\begin{aligned} \sin (x) + \sin (y) &= 2\sin ( \frac{x + y}{2} )\cos ( \frac{x - y}{2} ) \\ \sin (x) - \sin (y) &= 2\cos ( \frac{x + y}{2} )\sin ( \frac{x - y}{2} ) \\ \cos (x) + \cos (y) &= 2\cos ( \frac{x + y}{2} )\cos ( \frac{x - y}{2} ) \\ \cos (x) - \cos (y) &= - 2\sin ( \frac{x + y}{2} )\sin ( \frac{x - y}{2} ) \end{aligned}