Trigonometry Formulae

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### Basic Definitions - **SOH CAH TOA** (Right-angled triangles): - $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$ - $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$ - $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$ - **Reciprocal Identities:** - $\csc(\theta) = \frac{1}{\sin(\theta)}$ - $\sec(\theta) = \frac{1}{\cos(\theta)}$ - $\cot(\theta) = \frac{1}{\tan(\theta)}$ - **Quotient Identities:** - $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$ - $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$ ### Pythagorean Identities - $\sin^2(\theta) + \cos^2(\theta) = 1$ - $1 + \tan^2(\theta) = \sec^2(\theta)$ - $1 + \cot^2(\theta) = \csc^2(\theta)$ ### Angle Formulas #### Sum and Difference - $\sin(A \pm B) = \sin(A)\cos(B) \pm \cos(A)\sin(B)$ - $\cos(A \pm B) = \cos(A)\cos(B) \mp \sin(A)\sin(B)$ - $\tan(A \pm B) = \frac{\tan(A) \pm \tan(B)}{1 \mp \tan(A)\tan(B)}$ #### Double Angle - $\sin(2A) = 2\sin(A)\cos(A)$ - $\cos(2A) = \cos^2(A) - \sin^2(A)$ - $= 2\cos^2(A) - 1$ - $= 1 - 2\sin^2(A)$ - $\tan(2A) = \frac{2\tan(A)}{1 - \tan^2(A)}$ #### Half Angle - $\sin(\frac{A}{2}) = \pm\sqrt{\frac{1 - \cos(A)}{2}}$ - $\cos(\frac{A}{2}) = \pm\sqrt{\frac{1 + \cos(A)}{2}}$ - $\tan(\frac{A}{2}) = \frac{1 - \cos(A)}{\sin(A)} = \frac{\sin(A)}{1 + \cos(A)}$ #### Product-to-Sum - $2\sin(A)\cos(B) = \sin(A+B) + \sin(A-B)$ - $2\cos(A)\cos(B) = \cos(A+B) + \cos(A-B)$ - $2\sin(A)\sin(B) = \cos(A-B) - \cos(A+B)$ #### Sum-to-Product - $\sin(A) + \sin(B) = 2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)$ - $\sin(A) - \sin(B) = 2\cos\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right)$ - $\cos(A) + \cos(B) = 2\cos\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)$ - $\cos(A) - \cos(B) = -2\sin\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right)$ ### Law of Sines and Cosines #### Law of Sines - $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$ #### Law of Cosines - $c^2 = a^2 + b^2 - 2ab\cos(C)$ - $a^2 = b^2 + c^2 - 2bc\cos(A)$ - $b^2 = a^2 + c^2 - 2ac\cos(B)$ #### Area of a Triangle - Area $= \frac{1}{2}ab\sin(C) = \frac{1}{2}bc\sin(A) = \frac{1}{2}ac\sin(B)$ ### Unit Circle Values | Angle ($\theta$) | $\sin(\theta)$ | $\cos(\theta)$ | $\tan(\theta)$ | |:-----------------|:---------------|:---------------|:---------------| | $0^\circ$ (0 rad) | 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 | ### Periodicity and Co-function Identities #### Periodicity - $\sin(\theta + 2\pi k) = \sin(\theta)$ - $\cos(\theta + 2\pi k) = \cos(\theta)$ - $\tan(\theta + \pi k) = \tan(\theta)$ #### Co-function Identities - $\sin(\frac{\pi}{2} - \theta) = \cos(\theta)$ - $\cos(\frac{\pi}{2} - \theta) = \sin(\theta)$ - $\tan(\frac{\pi}{2} - \theta) = \cot(\theta)$