Trigonometry Identities

Cheatsheet Content

1. Basic Definitions For a right-angled triangle with angle $\theta$: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$ $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$ $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{\sin(\theta)}{\cos(\theta)}$ Reciprocal Identities: $\csc(\theta) = \frac{1}{\sin(\theta)}$ $\sec(\theta) = \frac{1}{\cos(\theta)}$ $\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)}$ 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. Angle Sum and Difference $\sin(A \pm B) = \sin(A)\cos(B) \pm \cos(A)\sin(B)$ $\cos(A \pm B) = \cos(A)\cos(B) \mp \sin(A)\sin(B)$ $\tan(A \pm B) = \frac{\tan(A) \pm \tan(B)}{1 \mp \tan(A)\tan(B)}$ 4. Double Angle Identities $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$ $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$ $\cos(2\theta) = 2\cos^2(\theta) - 1$ $\cos(2\theta) = 1 - 2\sin^2(\theta)$ $\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$ 5. Half Angle Identities $\sin(\frac{\theta}{2}) = \pm\sqrt{\frac{1 - \cos(\theta)}{2}}$ $\cos(\frac{\theta}{2}) = \pm\sqrt{\frac{1 + \cos(\theta)}{2}}$ $\tan(\frac{\theta}{2}) = \pm\sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}} = \frac{1 - \cos(\theta)}{\sin(\theta)} = \frac{\sin(\theta)}{1 + \cos(\theta)}$ 6. Product-to-Sum Identities $2\sin(A)\cos(B) = \sin(A+B) + \sin(A-B)$ $2\cos(A)\sin(B) = \sin(A+B) - \sin(A-B)$ $2\cos(A)\cos(B) = \cos(A+B) + \cos(A-B)$ $2\sin(A)\sin(B) = \cos(A-B) - \cos(A+B)$ 7. Sum-to-Product Identities $\sin(A) + \sin(B) = 2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)$ $\sin(A) - \sin(B) = 2\cos\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right)$ $\cos(A) + \cos(B) = 2\cos\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)$ $\cos(A) - \cos(B) = -2\sin\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right)$ 8. Cofunction Identities $\sin(\frac{\pi}{2} - \theta) = \cos(\theta)$ $\cos(\frac{\pi}{2} - \theta) = \sin(\theta)$ $\tan(\frac{\pi}{2} - \theta) = \cot(\theta)$ $\csc(\frac{\pi}{2} - \theta) = \sec(\theta)$ $\sec(\frac{\pi}{2} - \theta) = \csc(\theta)$ $\cot(\frac{\pi}{2} - \theta) = \tan(\theta)$ 9. Periodicity and Symmetry $\sin(\theta + 2\pi k) = \sin(\theta)$ $\cos(\theta + 2\pi k) = \cos(\theta)$ $\tan(\theta + \pi k) = \tan(\theta)$ $\sin(-\theta) = -\sin(\theta)$ (Odd function) $\cos(-\theta) = \cos(\theta)$ (Even function) $\tan(-\theta) = -\tan(\theta)$ (Odd function) 10. Law of Sines and Cosines (for any triangle) Law of Sines $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$ Law of Cosines $c^2 = a^2 + b^2 - 2ab\cos(C)$ $a^2 = b^2 + c^2 - 2bc\cos(A)$ $b^2 = a^2 + c^2 - 2ac\cos(B)$