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
$$
\begin{aligned}
\sin (-x) &= - \sin (x) \\
\cos (-x) &= - \cos (x) \\
\tan (-x) &= - \tan (x) \\
\csc (-x) &= - \csc (x) \\
\sec (-x) &= - \sec (x) \\
\cot (-x) &= - \cot (x)
\end{aligned}
$$
Cofunction Identity - radians
$$
\begin{aligned}
\sin ( \frac{\pi }{2} - x ) &= \cos (x) \\
\cos ( \frac{\pi }{2} - x ) &= \sin (x) \\
\tan ( \frac{\pi }{2} - x ) &= \cot (x) \\
\cot ( \frac{\pi }{2} - x ) &= \tan (x)
\end{aligned}
$$
Cofunction Identities - degrees
$$
\begin{aligned}
\sin ( 90^\circ - x ) &= \cos (x) \\
\cos ( 90^\circ - x ) &= \sin (x) \\
\tan ( 90^\circ - x ) &= \cot (x) \\
\cot ( 90^\circ - x ) &= \tan (x)
\end{aligned}
$$
Periodicity Identities - radians
$$
\begin{aligned}
\sin ( x + 2 \pi ) &= \sin (x) \\
\cos ( x + 2 \pi ) &= \cos (x) \\
\tan ( x + \pi ) &= \tan (x) \\
\cot ( x + \pi ) &= \cot (x)
\end{aligned}
$$
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
$$
\begin{aligned}
\sin (x) + \sin (y) &= 2\sin ( \frac{x + y}{2} )\cos ( \frac{x - y}{2} ) \\
\sin (x) - \sin (y) &= 2\cos ( \frac{x + y}{2} )\sin ( \frac{x - y}{2} ) \\
\cos (x) + \cos (y) &= 2\cos ( \frac{x + y}{2} )\cos ( \frac{x - y}{2} ) \\
\cos (x) - \cos (y) &= - 2\sin ( \frac{x + y}{2} )\sin ( \frac{x - y}{2} )
\end{aligned}
$$
