📐 Trigonometry
Mathematics | FirstInTest
| # | Trigonometry Formula | Notes | alternate formula |
|---|---|---|---|
| 1(i) | $\sin x = \frac{1}{\csc x}$ | — | $\csc x = \frac{1}{\sin x}$ |
| 1(ii) | $\cos x = \frac{1}{\sec x}$ | — | $\sec x = \frac{1}{\cos x}$ |
| 1(iii) | $\cot x = \frac{1}{\tan x}$ | — | $\tan x = \frac{1}{\cot x}$ |
| 1(iv) | $\tan x = \frac{\sin x}{\cos x}$ | — | $\cot x = \frac{\cos x}{\sin x}$ |
| 1(v) | $\sin^2 x + \cos^2 x = 1$ | — | — |
| 1(vi) | $1 + \tan^2 x = \sec^2 x$ | — | $\sec x - \tan x = \frac{1}{\sec x + \tan x}$ |
| 1(vii) | $1 + \cot^2 x = \csc^2 x$ | — | $\csc x - \cot x = \frac{1}{\csc x + \cot x}$ |
| 2(i) | $\sin (-x) = -\sin x$ | — | $\csc (-x) = -\csc x$ |
| 2(ii) | $\cos (-x) = \cos x$ | — | $\sec (-x) = \sec x$ |
| 2(iii) | $\tan (-x) = -\tan x$ | — | $\cot (-x) = -\cot x$ |
| 2(iv) | $\sin \left(\frac{\pi}{2} - x\right) = \cos x$ | Co-function identity | — |
| 2(iv) | $\cos \left(\frac{\pi}{2} - x\right) = \sin x$ | Co-function identity | — |
| 2(iv) | $\tan \left(\frac{\pi}{2} - x\right) = \cot x$ | Co-function identity | — |
| 2(iv) | $\sec \left(\frac{\pi}{2} - x\right) = \csc x$ | Co-function identity | — |
| 2(iv) | $\csc \left(\frac{\pi}{2} - x\right) = \sec x$ | Co-function identity | — |
| 2(iv) | $\cot \left(\frac{\pi}{2} - x\right) = \tan x$ | Co-function identity | — |
| 2(v) | $\sin \left(\frac{\pi}{2} + x\right) = \cos x$ | Quadrant II transformation | — |
| 2(v) | $\cos \left(\frac{\pi}{2} + x\right) = -\sin x$ | Quadrant II transformation | — |
| 2(v) | $\tan \left(\frac{\pi}{2} + x\right) = -\cot x$ | Quadrant II transformation | — |
| 2(v) | $\cot \left(\frac{\pi}{2} + x\right) = -\tan x$ | Quadrant II transformation | — |
| 2(v) | $\sec \left(\frac{\pi}{2} + x\right) = -\csc x$ | Quadrant II transformation | — |
| 2(v) | $\csc \left(\frac{\pi}{2} + x\right) = \sec x$ | Quadrant II transformation | — |
| 2(vi) | $\sin (\pi - x) = \sin x$ | Quadrant II transformation | — |
| 2(vi) | $\cos (\pi - x) = -\cos x$ | Quadrant II transformation | — |
| 2(vi) | $\tan (\pi - x) = -\tan x$ | Quadrant II transformation | — |
| 2(vi) | $\cot (\pi - x) = -\cot x$ | Quadrant II transformation | — |
| 2(vi) | $\sec (\pi - x) = -\sec x$ | Quadrant II transformation | — |
| 2(vi) | $\csc (\pi - x) = \csc x$ | Quadrant II transformation | — |
| 2(vii) | $\sin \left(\frac{3\pi}{2} - x\right) = -\cos x$ | Quadrant III transformation | — |
| 2(vii) | $\cos \left(\frac{3\pi}{2} - x\right) = -\sin x$ | Quadrant III transformation | — |
| 2(vii) | $\tan \left(\frac{3\pi}{2} - x\right) = \cot x$ | Quadrant III transformation | — |
| 2(vii) | $\cot \left(\frac{3\pi}{2} - x\right) = \tan x$ | Quadrant III transformation | — |
| 2(vii) | $\csc \left(\frac{3\pi}{2} - x\right) = -\sec x$ | Quadrant III transformation | — |
| 2(vii) | $\sec \left(\frac{3\pi}{2} - x\right) = -\csc x$ | Quadrant III transformation | — |
| 2(viii) | $\sin \left(\frac{3\pi}{2} + x\right) = -\cos x$ | Quadrant IV transformation | — |
| 2(viii) | $\cos \left(\frac{3\pi}{2} + x\right) = \sin x$ | Quadrant IV transformation | — |
| 2(viii) | $\tan \left(\frac{3\pi}{2} + x\right) = -\cot x$ | Quadrant IV transformation | — |
| 2(viii) | $\cot \left(\frac{3\pi}{2} + x\right) = -\tan x$ | Quadrant IV transformation | — |
| 2(viii) | $\csc \left(\frac{3\pi}{2} + x\right) = -\sec x$ | Quadrant IV transformation | — |
| 2(viii) | $\sec \left(\frac{3\pi}{2} + x\right) = \csc x$ | Quadrant IV transformation | — |
| 2(ix) | $\sin (2\pi - x) = -\sin x$ | Quadrant IV transformation (periodic function) | — |
| 2(ix) | $\cos (2\pi - x) = \cos x$ | Quadrant IV transformation (periodic function) | — |
| 2(ix) | $\tan (2\pi - x) = -\tan x$ | Quadrant IV transformation (periodic function) | — |
| 2(ix) | $\csc (2\pi - x) = -\csc x$ | Quadrant IV transformation (periodic function) | — |
| 2(ix) | $\sec (2\pi - x) = \sec x$ | Quadrant IV transformation (periodic function) | — |
| 2(ix) | $\cot (2\pi - x) = -\cot x$ | Quadrant IV transformation (periodic function) | — |
| 2(x) | Sine, Cosine, Cosecant, Secant functions are periodic with period $2\pi$. | — | Tangent and Cotangent functions are periodic with period $\pi$. |
| 2(xi) | Cosine and Secant functions are even functions. | — | Sine, Cosecant, Tangent, Cotangent functions are odd functions. |
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