Trigonometry All Formulae

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I. Basic Identities Reciprocal Identities: $\sin \theta = \frac{1}{\csc \theta}$ $\cos \theta = \frac{1}{\sec \theta}$ $\tan \theta = \frac{1}{\cot \theta}$ $\csc \theta = \frac{1}{\sin \theta}$ $\sec \theta = \frac{1}{\cos \theta}$ $\cot \theta = \frac{1}{\tan \theta}$ Quotient Identities: $\tan \theta = \frac{\sin \theta}{\cos \theta}$ $\cot \theta = \frac{\cos \theta}{\sin \theta}$ Pythagorean Identities: $\sin^2 \theta + \cos^2 \theta = 1$ $1 + \tan^2 \theta = \sec^2 \theta$ $1 + \cot^2 \theta = \csc^2 \theta$ II. Angle Sum and Difference Formulas $\sin(A+B) = \sin A \cos B + \cos A \sin B$ $\sin(A-B) = \sin A \cos B - \cos A \sin B$ $\cos(A+B) = \cos A \cos B - \sin A \sin B$ $\cos(A-B) = \cos A \cos B + \sin A \sin B$ $\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$ $\tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$ III. Double Angle Formulas $\sin(2A) = 2 \sin A \cos A$ $\cos(2A) = \cos^2 A - \sin^2 A$ $\cos(2A) = 2 \cos^2 A - 1$ $\cos(2A) = 1 - 2 \sin^2 A$ $\tan(2A) = \frac{2 \tan A}{1 - \tan^2 A}$ IV. Half Angle Formulas $\sin\left(\frac{A}{2}\right) = \pm \sqrt{\frac{1 - \cos A}{2}}$ $\cos\left(\frac{A}{2}\right) = \pm \sqrt{\frac{1 + \cos A}{2}}$ $\tan\left(\frac{A}{2}\right) = \pm \sqrt{\frac{1 - \cos A}{1 + \cos A}}$ $\tan\left(\frac{A}{2}\right) = \frac{1 - \cos A}{\sin A}$ $\tan\left(\frac{A}{2}\right) = \frac{\sin A}{1 + \cos A}$ V. Product-to-Sum Formulas $2 \sin A \cos B = \sin(A+B) + \sin(A-B)$ $2 \cos A \sin B = \sin(A+B) - \sin(A-B)$ $2 \cos A \cos B = \cos(A+B) + \cos(A-B)$ $2 \sin A \sin B = \cos(A-B) - \cos(A+B)$ VI. Sum-to-Product Formulas $\sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right)$ $\sin A - \sin B = 2 \cos\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right)$ $\cos A + \cos B = 2 \cos\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right)$ $\cos A - \cos B = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right)$ VII. Cofunction Identities $\sin\left(\frac{\pi}{2} - \theta\right) = \cos \theta$ $\cos\left(\frac{\pi}{2} - \theta\right) = \sin \theta$ $\tan\left(\frac{\pi}{2} - \theta\right) = \cot \theta$ $\csc\left(\frac{\pi}{2} - \theta\right) = \sec \theta$ $\sec\left(\frac{\pi}{2} - \theta\right) = \csc \theta$ $\cot\left(\frac{\pi}{2} - \theta\right) = \tan \theta$ VIII. Periodicity Identities $\sin(\theta + 2\pi n) = \sin \theta$ $\cos(\theta + 2\pi n) = \cos \theta$ $\tan(\theta + \pi n) = \tan \theta$ IX. Even/Odd Identities $\sin(-\theta) = -\sin \theta$ (Odd) $\cos(-\theta) = \cos \theta$ (Even) $\tan(-\theta) = -\tan \theta$ (Odd) X. Inverse Trigonometric Functions $\arcsin x$ domain: $[-1, 1]$, range: $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ $\arccos x$ domain: $[-1, 1]$, range: $[0, \pi]$ $\arctan x$ domain: $(-\infty, \infty)$, range: $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ $\arcsin x + \arccos x = \frac{\pi}{2}$ $\arctan x + \text{arccot } x = \frac{\pi}{2}$ $\text{arcsec } x + \text{arccsc } x = \frac{\pi}{2}$ XI. Law of Sines and Cosines (for any triangle ABC) Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ Law of Cosines: $c^2 = a^2 + b^2 - 2ab \cos C$ $b^2 = a^2 + c^2 - 2ac \cos B$ $a^2 = b^2 + c^2 - 2bc \cos A$ XII. Area of a Triangle Area $= \frac{1}{2}ab \sin C = \frac{1}{2}bc \sin A = \frac{1}{2}ca \sin B$ Heron's Formula: Area $= \sqrt{s(s-a)(s-b)(s-c)}$, where $s = \frac{a+b+c}{2}$ (semi-perimeter) XIII. Important Values $\theta$ (degrees) $\theta$ (radians) $\sin \theta$ $\cos \theta$ $\tan \theta$ $0^\circ$ $0$ $0$ $1$ $0$ $30^\circ$ $\frac{\pi}{6}$ $\frac{1}{2}$ $\frac{\sqrt{3}}{2}$ $\frac{1}{\sqrt{3}}$ $45^\circ$ $\frac{\pi}{4}$ $\frac{\sqrt{2}}{2}$ $\frac{\sqrt{2}}{2}$ $1$ $60^\circ$ $\frac{\pi}{3}$ $\frac{\sqrt{3}}{2}$ $\frac{1}{2}$ $\sqrt{3}$ $90^\circ$ $\frac{\pi}{2}$ $1$ $0$ Undefined $180^\circ$ $\pi$ $0$ $-1$ $0$ $270^\circ$ $\frac{3\pi}{2}$ $-1$ $0$ Undefined $360^\circ$ $2\pi$ $0$ $1$ $0$