2019-11-03_20-47-06

Trigonometric Identities

\cos^2 x + \sin^2 x = 1

1 + \tan^2 x = \sec^2 x for all x \in R - \Big\{ (2n-1) \frac{\pi}{2} : n \in Z \Big\}

1 + \cot^2 x = \mathrm{cpsec}^2 x for all  x \in R - \{ n \pi : n \in Z \}

Trigonometric Functions in Different Quadrants

\theta lies in Quadrant II

\sin \theta : +ve      \mathrm{cosec} \theta : +ve

\cos \theta : -ve      \sec \theta : -ve

\tan \theta : -ve      \cot \theta : -ve

\theta lies in Quadrant I

\sin \theta : +ve      \mathrm{cosec} \theta : +ve

\cos \theta : +ve      \sec \theta : +ve

\tan \theta : +ve      \cot \theta : +ve

\theta lies in Quadrant III

\sin \theta : -ve      \mathrm{cosec} \theta : -ve

\cos \theta : -ve      \sec \theta : -ve

\tan \theta : +ve      \cot \theta : +ve

\theta lies in Quadrant IV

\sin \theta : -ve      \mathrm{cosec} \theta : -ve

\cos \theta : +ve      \sec \theta : +ve

\tan \theta : -ve      \cot \theta : -ve

The value of the trigonometric functions are given in terms of x in the table below.

2019-11-03_21-48-11.png \sin \cos \tan \mathrm{cosec} x \sec

\cot

-x

- \sin x  \cos x - \tan x -\mathrm{cosec} x \sec x

-\cot x

\frac{\pi}{2} - x

 \cos x \sin x \cot x \sec x \mathrm{cosec} x

\tan x

\frac{\pi}{2} + x

 \cos x - \sin x - \cot x \sec x -\mathrm{cosec} x

-\tan x

\pi - x

 \sin x -\cos x - \tan x \mathrm{cosec} x -\sec x

-\cot x

\pi + x

- \sin x -\cos x \tan x -\mathrm{cosec} x -\sec x

\cot x

\frac{3\pi}{2} - x

-\cos x - \sin x \cot x -\sec x -\mathrm{cosec} x

\tan x

\frac{3\pi}{2} + x

-\cos x \sin x - \cot x -\sec x \mathrm{cosec} x

-\tan x

2\pi - x

- \sin x  \cos x - \tan x -\mathrm{cosec} x \sec x

-\cot x

2\pi + x \sin x  \cos x \tan x \mathrm{cosec} x \sec x

\cot x