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03Class 11 Maths

Trigonometric Functions formulas

The trigonometric identities and formulas for Class 11. Trigonometry returns in almost every Maths chapter that follows, so this sheet is worth learning cold.

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All Maths formulas

Basic identities

Pythagorean identities
sin⁡2x+cos⁡2x=1,1+tan⁡2x=sec⁡2x,1+cot⁡2x=csc⁡2x\displaystyle \sin^2x + \cos^2x = 1, \quad 1 + \tan^2x = \sec^2x, \quad 1 + \cot^2x = \csc^2x
Radians and degrees
π rad=180∘\displaystyle \pi\ \text{rad} = 180^\circ
Arc length
l=rθ\displaystyle l = r\theta

θ in radians.

Sum and difference

Sine
sin⁡(x±y)=sin⁡xcos⁡y±cos⁡xsin⁡y\displaystyle \sin(x \pm y) = \sin x\cos y \pm \cos x\sin y
Cosine
cos⁡(x±y)=cos⁡xcos⁡y∓sin⁡xsin⁡y\displaystyle \cos(x \pm y) = \cos x\cos y \mp \sin x\sin y
Tangent
tan⁡(x±y)=tan⁡x±tan⁡y1∓tan⁡xtan⁡y\displaystyle \tan(x \pm y) = \frac{\tan x \pm \tan y}{1 \mp \tan x\tan y}

Double and triple angle

sin 2x
sin⁡2x=2sin⁡xcos⁡x=2tan⁡x1+tan⁡2x\displaystyle \sin 2x = 2\sin x\cos x = \frac{2\tan x}{1 + \tan^2x}
cos 2x
cos⁡2x=cos⁡2x−sin⁡2x=2cos⁡2x−1=1−2sin⁡2x\displaystyle \cos 2x = \cos^2x - \sin^2x = 2\cos^2x - 1 = 1 - 2\sin^2x
tan 2x
tan⁡2x=2tan⁡x1−tan⁡2x\displaystyle \tan 2x = \frac{2\tan x}{1 - \tan^2x}
Triple angle
sin⁡3x=3sin⁡x−4sin⁡3x,cos⁡3x=4cos⁡3x−3cos⁡x\displaystyle \sin 3x = 3\sin x - 4\sin^3x, \quad \cos 3x = 4\cos^3x - 3\cos x

Sum to product

Sines
sin⁡x+sin⁡y=2sin⁡x+y2cos⁡x−y2\displaystyle \sin x + \sin y = 2\sin\frac{x + y}{2}\cos\frac{x - y}{2}
Cosines
cos⁡x+cos⁡y=2cos⁡x+y2cos⁡x−y2\displaystyle \cos x + \cos y = 2\cos\frac{x + y}{2}\cos\frac{x - y}{2}
Difference of cosines
cos⁡x−cos⁡y=−2sin⁡x+y2sin⁡x−y2\displaystyle \cos x - \cos y = -2\sin\frac{x + y}{2}\sin\frac{x - y}{2}

General solutions

sin x = 0 and cos x = 0
sin⁡x=0⇒x=nπ,cos⁡x=0⇒x=(2n+1)π2\displaystyle \sin x = 0 \Rightarrow x = n\pi, \quad \cos x = 0 \Rightarrow x = (2n + 1)\frac{\pi}{2}
sin x = sin y
x=nπ+(−1)ny\displaystyle x = n\pi + (-1)^n y
cos x = cos y
x=2nπ±y\displaystyle x = 2n\pi \pm y
tan x = tan y
x=nπ+y\displaystyle x = n\pi + y

Where students slip up

  • Forgetting the sign flip in cos(x + y), which uses a minus between its terms.
  • Cancelling sin x from both sides of an equation. You lose the solutions where sin x = 0.
  • Giving one angle when the question asks for the general solution.

Try it on your weakest chapter.

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