Trigonometric Ratios Cheatsheet

Cheatsheet Content

1. Right-Angled Triangle Basics Hypotenuse (H): Side opposite the right angle. Opposite (O): Side opposite to the angle $\theta$. Adjacent (A): Side next to the angle $\theta$ (not the hypotenuse). Pythagorean Theorem: $O^2 + A^2 = H^2$ 2. Basic Trigonometric Ratios (SOH CAH TOA) Sine ($\sin \theta$): Ratio of the length of the opposite side to the length of the hypotenuse. $\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{O}{H}$ Cosine ($\cos \theta$): Ratio of the length of the adjacent side to the length of the hypotenuse. $\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{A}{H}$ Tangent ($\tan \theta$): Ratio of the length of the opposite side to the length of the adjacent side. $\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{O}{A}$ 3. Reciprocal Trigonometric Ratios Cosecant ($\csc \theta$ or $\text{cosec } \theta$): Reciprocal of sine. $\csc \theta = \frac{1}{\sin \theta} = \frac{H}{O}$ Secant ($\sec \theta$): Reciprocal of cosine. $\sec \theta = \frac{1}{\cos \theta} = \frac{H}{A}$ Cotangent ($\cot \theta$ or $\text{cotan } \theta$): Reciprocal of tangent. $\cot \theta = \frac{1}{\tan \theta} = \frac{A}{O}$ 4. Quotient Identities $\tan \theta = \frac{\sin \theta}{\cos \theta}$ $\cot \theta = \frac{\cos \theta}{\sin \theta}$ 5. Pythagorean Identities $\sin^2 \theta + \cos^2 \theta = 1$ $1 + \tan^2 \theta = \sec^2 \theta$ $1 + \cot^2 \theta = \csc^2 \theta$ 6. Important Angle Values (Degrees and Radians) $\theta$ $0^\circ$ (0) $30^\circ$ ($\pi/6$) $45^\circ$ ($\pi/4$) $60^\circ$ ($\pi/3$) $90^\circ$ ($\pi/2$) $180^\circ$ ($\pi$) $270^\circ$ ($3\pi/2$) $360^\circ$ ($2\pi$) $\sin \theta$ 0 $1/2$ $\sqrt{2}/2$ $\sqrt{3}/2$ 1 0 -1 0 $\cos \theta$ 1 $\sqrt{3}/2$ $\sqrt{2}/2$ $1/2$ 0 -1 0 1 $\tan \theta$ 0 $1/\sqrt{3}$ 1 $\sqrt{3}$ Undefined 0 Undefined 0 7. Angle Sum and Difference Formulas $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$ $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$ $\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$ 8. Double Angle Formulas $\sin(2A) = 2 \sin A \cos A$ $\cos(2A) = \cos^2 A - \sin^2 A$ $\cos(2A) = 2 \cos^2 A - 1$ $\cos(2A) = 1 - 2 \sin^2 A$ $\tan(2A) = \frac{2 \tan A}{1 - \tan^2 A}$ 9. Half Angle Formulas $\sin(\frac{A}{2}) = \pm \sqrt{\frac{1 - \cos A}{2}}$ $\cos(\frac{A}{2}) = \pm \sqrt{\frac{1 + \cos A}{2}}$ $\tan(\frac{A}{2}) = \pm \sqrt{\frac{1 - \cos A}{1 + \cos A}}$ $\tan(\frac{A}{2}) = \frac{1 - \cos A}{\sin A} = \frac{\sin A}{1 + \cos A}$ 10. Product-to-Sum Formulas $2 \sin A \cos B = \sin(A+B) + \sin(A-B)$ $2 \cos A \sin B = \sin(A+B) - \sin(A-B)$ $2 \cos A \cos B = \cos(A+B) + \cos(A-B)$ $2 \sin A \sin B = \cos(A-B) - \cos(A+B)$ 11. Sum-to-Product Formulas $\sin A + \sin B = 2 \sin \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)$ $\sin A - \sin B = 2 \cos \left(\frac{A+B}{2}\right) \sin \left(\frac{A-B}{2}\right)$ $\cos A + \cos B = 2 \cos \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)$ $\cos A - \cos B = -2 \sin \left(\frac{A+B}{2}\right) \sin \left(\frac{A-B}{2}\right)$ 12. Cofunction Identities $\sin(\frac{\pi}{2} - \theta) = \cos \theta$ $\cos(\frac{\pi}{2} - \theta) = \sin \theta$ $\tan(\frac{\pi}{2} - \theta) = \cot \theta$ $\csc(\frac{\pi}{2} - \theta) = \sec \theta$ $\sec(\frac{\pi}{2} - \theta) = \csc \theta$ $\cot(\frac{\pi}{2} - \theta) = \tan \theta$ 13. Periodicity $\sin(\theta + 2\pi k) = \sin \theta$ $\cos(\theta + 2\pi k) = \cos \theta$ $\tan(\theta + \pi k) = \tan \theta$ (where $k$ is any integer) 14. General Solutions of Trigonometric Equations If $\sin \theta = \sin \alpha$, then $\theta = n\pi + (-1)^n \alpha$, where $n \in \mathbb{Z}$. If $\cos \theta = \cos \alpha$, then $\theta = 2n\pi \pm \alpha$, where $n \in \mathbb{Z}$. If $\tan \theta = \tan \alpha$, then $\theta = n\pi + \alpha$, where $n \in \mathbb{Z}$.