Chapters

Trigonometric Functions

NCERT Class 11 Mathematics — Chapter 3

Trigonometry is a smuggler, a free-trader in mathematics. — Augustus De Morgan

3.1 Angles

An angle is formed by rotating a ray about its endpoint. Anticlockwise is positive.

Degree and radian:

π radians=180∘,1∘=π180 rad\pi\text{ radians}=180^\circ,\qquad 1^\circ=\frac{\pi}{180}\text{ rad}

A full rotation is 2π2\pi radians.

3.2 Trigonometric Functions

For a real number xx (radians), sin⁡x\sin x, cos⁡x\cos x, tan⁡x=sin⁡x/cos⁡x\tan x=\sin x/\cos x (where cos⁡x≠0\cos x\mathrel{\htmlClass{course-neq}{\char"2260}} 0), and the reciprocals csc⁡\csc, sec⁡\sec, cot⁡\cot are defined using the unit circle (or wrapping of the real line).

Signs follow the quadrant of the terminal side.

Standard values (selected)

sin⁡π6=12,sin⁡π4=12,sin⁡π3=32,sin⁡π2=1\sin\frac{\pi}{6}=\frac12,\quad \sin\frac{\pi}{4}=\frac{1}{\sqrt{2}},\quad \sin\frac{\pi}{3}=\frac{\sqrt{3}}{2},\quad \sin\frac{\pi}{2}=1

Cosine of the same angles: 3/2\sqrt{3}/2, 1/21/\sqrt{2}, 1/21/2, 00.

3.3 Identities

sin⁡2x+cos⁡2x=1\sin^2 x+\cos^2 x=1

1+tan⁡2x=sec⁡2x,1+cot⁡2x=csc⁡2x1+\tan^2 x=\sec^2 x,\qquad 1+\cot^2 x=\csc^2 x

Compound angles:

sin⁡(x±y)=sin⁡xcos⁡y±cos⁡xsin⁡y\sin(x\pm y)=\sin x\cos y\pm\cos x\sin y

cos⁡(x±y)=cos⁡xcos⁡y∓sin⁡xsin⁡y\cos(x\pm y)=\cos x\cos y\mp\sin x\sin y

tan⁡(x±y)=tan⁡x±tan⁡y1∓tan⁡xtan⁡y\tan(x\pm y)=\frac{\tan x\pm\tan y}{1\mp\tan x\tan y}

Double angle: sin⁡2x=2sin⁡xcos⁡x\sin 2x=2\sin x\cos x, cos⁡2x=cos⁡2x−sin⁡2x\cos 2x=\cos^2 x-\sin^2 x.

Chapter Summary

  • Radian measure: π\pi rad =180∘=180^\circ
  • sin⁡,cos⁡,tan⁡\sin,\cos,\tan and reciprocals as functions of a real variable
  • Pythagorean identities; compound-angle and double-angle formulae

Exercises (NCERT)

  • Exercise 3.1 — degree/radian
  • Exercise 3.2 onwards — values and identities
  • Miscellaneous Exercise