Class 11 CBSE Trigonometry

Cheatsheet Content

1. Angles and Their Measurement Degree Measure: $1^\circ = 60'$, $1' = 60''$ Radian Measure: Angle subtended at the center by an arc of length equal to the radius. Relation: $\pi \text{ radians} = 180^\circ$ $1 \text{ radian} = \frac{180^\circ}{\pi} \approx 57^\circ 16'$ $1^\circ = \frac{\pi}{180} \text{ radians} \approx 0.01746 \text{ radians}$ Arc Length: $l = r\theta$, where $\theta$ is in radians. 2. Trigonometric Ratios/Functions Basic Ratios (Right Triangle) $\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$ $\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$ $\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$ $\csc \theta = \frac{1}{\sin \theta}$ $\sec \theta = \frac{1}{\cos \theta}$ $\cot \theta = \frac{1}{\tan \theta}$ Signs of Trigonometric Functions Quadrant I II III IV $\sin \theta$ + + - - $\cos \theta$ + - - + $\tan \theta$ + - + - "All Students Take Coffee" rule. 3. Trigonometric Identities $\sin^2 \theta + \cos^2 \theta = 1$ $1 + \tan^2 \theta = \sec^2 \theta$ $1 + \cot^2 \theta = \csc^2 \theta$ 4. Trigonometric Functions of Allied Angles $\sin(2n\pi + x) = \sin x$ $\cos(2n\pi + x) = \cos x$ $\sin(-x) = -\sin x$ $\cos(-x) = \cos x$ $\tan(-x) = -\tan x$ $\sin(\frac{\pi}{2} - x) = \cos x$ $\cos(\frac{\pi}{2} - x) = \sin x$ $\tan(\frac{\pi}{2} - x) = \cot x$ $\sin(\frac{\pi}{2} + x) = \cos x$ $\cos(\frac{\pi}{2} + x) = -\sin x$ $\tan(\frac{\pi}{2} + x) = -\cot x$ $\sin(\pi - x) = \sin x$ $\cos(\pi - x) = -\cos x$ $\tan(\pi - x) = -\tan x$ $\sin(\pi + x) = -\sin x$ $\cos(\pi + x) = -\cos x$ $\tan(\pi + x) = \tan x$ $\sin(\frac{3\pi}{2} - x) = -\cos x$ $\cos(\frac{3\pi}{2} - x) = -\sin x$ $\tan(\frac{3\pi}{2} - x) = \cot x$ $\sin(\frac{3\pi}{2} + x) = -\cos x$ $\cos(\frac{3\pi}{2} + x) = \sin x$ $\tan(\frac{3\pi}{2} + x) = -\cot x$ $\sin(2\pi - x) = -\sin x$ $\cos(2\pi - x) = \cos x$ $\tan(2\pi - x) = -\tan x$ 5. Compound Angles $\sin(x+y) = \sin x \cos y + \cos x \sin y$ $\sin(x-y) = \sin x \cos y - \cos x \sin y$ $\cos(x+y) = \cos x \cos y - \sin x \sin y$ $\cos(x-y) = \cos x \cos y + \sin x \sin y$ $\tan(x+y) = \frac{\tan x + \tan y}{1 - \tan x \tan y}$ $\tan(x-y) = \frac{\tan x - \tan y}{1 + \tan x \tan y}$ $\cot(x+y) = \frac{\cot x \cot y - 1}{\cot y + \cot x}$ $\cot(x-y) = \frac{\cot x \cot y + 1}{\cot y - \cot x}$ 6. Multiple and Submultiple Angles Double Angle Formulas $\sin 2x = 2 \sin x \cos x = \frac{2 \tan x}{1 + \tan^2 x}$ $\cos 2x = \cos^2 x - \sin^2 x = 2 \cos^2 x - 1 = 1 - 2 \sin^2 x = \frac{1 - \tan^2 x}{1 + \tan^2 x}$ $\tan 2x = \frac{2 \tan x}{1 - \tan^2 x}$ $1 + \cos 2x = 2 \cos^2 x$ $1 - \cos 2x = 2 \sin^2 x$ Triple Angle Formulas $\sin 3x = 3 \sin x - 4 \sin^3 x$ $\cos 3x = 4 \cos^3 x - 3 \cos x$ $\tan 3x = \frac{3 \tan x - \tan^3 x}{1 - 3 \tan^2 x}$ Half Angle Formulas (from Double Angle) $\sin x = 2 \sin \frac{x}{2} \cos \frac{x}{2}$ $\cos x = \cos^2 \frac{x}{2} - \sin^2 \frac{x}{2}$ $\tan x = \frac{2 \tan \frac{x}{2}}{1 - \tan^2 \frac{x}{2}}$ $\sin^2 \frac{x}{2} = \frac{1 - \cos x}{2}$ $\cos^2 \frac{x}{2} = \frac{1 + \cos x}{2}$ $\tan^2 \frac{x}{2} = \frac{1 - \cos x}{1 + \cos x}$ 7. Transformation Formulas (Product to Sum/Difference) $2 \sin A \cos B = \sin(A+B) + \sin(A-B)$ $2 \cos A \sin B = \sin(A+B) - \sin(A-B)$ $2 \cos A \cos B = \cos(A+B) + \cos(A-B)$ $2 \sin A \sin B = \cos(A-B) - \cos(A+B)$ 8. Transformation Formulas (Sum/Difference to Product) $\sin C + \sin D = 2 \sin \frac{C+D}{2} \cos \frac{C-D}{2}$ $\sin C - \sin D = 2 \cos \frac{C+D}{2} \sin \frac{C-D}{2}$ $\cos C + \cos D = 2 \cos \frac{C+D}{2} \cos \frac{C-D}{2}$ $\cos C - \cos D = -2 \sin \frac{C+D}{2} \sin \frac{C-D}{2}$ 9. General Solutions of Trigonometric Equations If $\sin x = \sin y$, then $x = n\pi + (-1)^n y$, where $n \in \mathbb{Z}$. If $\cos x = \cos y$, then $x = 2n\pi \pm y$, where $n \in \mathbb{Z}$. If $\tan x = \tan y$, then $x = n\pi + y$, where $n \in \mathbb{Z}$. If $\sin^2 x = \sin^2 y$ or $\cos^2 x = \cos^2 y$ or $\tan^2 x = \tan^2 y$, then $x = n\pi \pm y$, where $n \in \mathbb{Z}$. 10. Values of Trigonometric Functions Angle ($\theta$) $0^\circ$ (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 $1/2$ $1/\sqrt{2}$ $\sqrt{3}/2$ 1 0 -1 0 $\cos \theta$ 1 $\sqrt{3}/2$ $1/\sqrt{2}$ $1/2$ 0 -1 0 1 $\tan \theta$ 0 $1/\sqrt{3}$ 1 $\sqrt{3}$ Undefined 0 Undefined 0 Angle ($\theta$) $15^\circ$ ($\pi/12$) $75^\circ$ ($5\pi/12$) $\sin \theta$ $\frac{\sqrt{3}-1}{2\sqrt{2}}$ $\frac{\sqrt{3}+1}{2\sqrt{2}}$ $\cos \theta$ $\frac{\sqrt{3}+1}{2\sqrt{2}}$ $\frac{\sqrt{3}-1}{2\sqrt{2}}$ $\tan \theta$ $2-\sqrt{3}$ $2+\sqrt{3}$