Trigonometry Formulas Cheatsheet

Cheatsheet Content

1. Basic Definitions (Right Triangle) $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$ $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$ $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{\sin(\theta)}{\cos(\theta)}$ $\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{\text{Hypotenuse}}{\text{Opposite}}$ $\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\text{Hypotenuse}}{\text{Adjacent}}$ $\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)}$ 2. Pythagorean Identities $\sin^2(\theta) + \cos^2(\theta) = 1$ $1 + \tan^2(\theta) = \sec^2(\theta)$ $1 + \cot^2(\theta) = \csc^2(\theta)$ 3. Reciprocal & Quotient Identities $\sin(\theta) = \frac{1}{\csc(\theta)}$ $\cos(\theta) = \frac{1}{\sec(\theta)}$ $\tan(\theta) = \frac{1}{\cot(\theta)}$ $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$ $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$ 4. Cofunction Identities $\sin(\frac{\pi}{2} - \theta) = \cos(\theta)$ $\cos(\frac{\pi}{2} - \theta) = \sin(\theta)$ $\tan(\frac{\pi}{2} - \theta) = \cot(\theta)$ $\csc(\frac{\pi}{2} - \theta) = \sec(\theta)$ $\sec(\frac{\pi}{2} - \theta) = \csc(\theta)$ $\cot(\frac{\pi}{2} - \theta) = \tan(\theta)$ 5. Even/Odd Identities $\sin(-\theta) = -\sin(\theta)$ (Odd) $\cos(-\theta) = \cos(\theta)$ (Even) $\tan(-\theta) = -\tan(\theta)$ (Odd) $\csc(-\theta) = -\csc(\theta)$ (Odd) $\sec(-\theta) = \sec(\theta)$ (Even) $\cot(-\theta) = -\cot(\theta)$ (Odd) 6. Sum and Difference Formulas $\sin(A \pm B) = \sin(A)\cos(B) \pm \cos(A)\sin(B)$ $\cos(A \pm B) = \cos(A)\cos(B) \mp \sin(A)\sin(B)$ $\tan(A \pm B) = \frac{\tan(A) \pm \tan(B)}{1 \mp \tan(A)\tan(B)}$ 7. Double Angle Formulas $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$ $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$ $\cos(2\theta) = 2\cos^2(\theta) - 1$ $\cos(2\theta) = 1 - 2\sin^2(\theta)$ $\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$ 8. Half Angle Formulas $\sin(\frac{\theta}{2}) = \pm\sqrt{\frac{1 - \cos(\theta)}{2}}$ $\cos(\frac{\theta}{2}) = \pm\sqrt{\frac{1 + \cos(\theta)}{2}}$ $\tan(\frac{\theta}{2}) = \pm\sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}}$ $\tan(\frac{\theta}{2}) = \frac{1 - \cos(\theta)}{\sin(\theta)}$ $\tan(\frac{\theta}{2}) = \frac{\sin(\theta)}{1 + \cos(\theta)}$ 9. Product-to-Sum Formulas $2\sin(A)\cos(B) = \sin(A+B) + \sin(A-B)$ $2\cos(A)\sin(B) = \sin(A+B) - \sin(A-B)$ $2\cos(A)\cos(B) = \cos(A+B) + \cos(A-B)$ $2\sin(A)\sin(B) = \cos(A-B) - \cos(A+B)$ 10. Sum-to-Product Formulas $\sin(A) + \sin(B) = 2\sin(\frac{A+B}{2})\cos(\frac{A-B}{2})$ $\sin(A) - \sin(B) = 2\cos(\frac{A+B}{2})\sin(\frac{A-B}{2})$ $\cos(A) + \cos(B) = 2\cos(\frac{A+B}{2})\cos(\frac{A-B}{2})$ $\cos(A) - \cos(B) = -2\sin(\frac{A+B}{2})\sin(\frac{A-B}{2})$ 11. Law of Sines For any triangle with sides $a, b, c$ and opposite angles $A, B, C$: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$ 12. Law of Cosines $c^2 = a^2 + b^2 - 2ab\cos(C)$ $a^2 = b^2 + c^2 - 2bc\cos(A)$ $b^2 = a^2 + c^2 - 2ac\cos(B)$ 13. Area of a Triangle Area $= \frac{1}{2}ab\sin(C)$ Area $= \frac{1}{2}bc\sin(A)$ Area $= \frac{1}{2}ac\sin(B)$ Heron's Formula: Area $= \sqrt{s(s-a)(s-b)(s-c)}$, where $s = \frac{a+b+c}{2}$ 14. Inverse Trigonometric Functions $\arcsin(x)$ or $\sin^{-1}(x)$: range $[-\frac{\pi}{2}, \frac{\pi}{2}]$ $\arccos(x)$ or $\cos^{-1}(x)$: range $[0, \pi]$ $\arctan(x)$ or $\tan^{-1}(x)$: range $(-\frac{\pi}{2}, \frac{\pi}{2})$ Inverse Identities $\sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2}$ $\tan^{-1}(x) + \cot^{-1}(x) = \frac{\pi}{2}$ $\sec^{-1}(x) + \csc^{-1}(x) = \frac{\pi}{2}$ $\tan^{-1}(x) + \tan^{-1}(y) = \tan^{-1}(\frac{x+y}{1-xy})$ 15. Unit Circle Values (Common Angles) Angle ($\theta$) $\sin(\theta)$ $\cos(\theta)$ $\tan(\theta)$ $0$ ($0^\circ$) $0$ $1$ $0$ $\frac{\pi}{6}$ ($30^\circ$) $\frac{1}{2}$ $\frac{\sqrt{3}}{2}$ $\frac{1}{\sqrt{3}}$ $\frac{\pi}{4}$ ($45^\circ$) $\frac{\sqrt{2}}{2}$ $\frac{\sqrt{2}}{2}$ $1$ $\frac{\pi}{3}$ ($60^\circ$) $\frac{\sqrt{3}}{2}$ $\frac{1}{2}$ $\sqrt{3}$ $\frac{\pi}{2}$ ($90^\circ$) $1$ $0$ Undefined $\pi$ ($180^\circ$) $0$ $-1$ $0$ $\frac{3\pi}{2}$ ($270^\circ$) $-1$ $0$ Undefined $2\pi$ ($360^\circ$) $0$ $1$ $0$