Trigonometry

high importance~3 Q in Tier 119 formulas⚡ 11 shortcuts5 subtopics

Every formula in this topic, grouped by subtopic. Print it and pin it above your desk.

Ratios and standard values

Reciprocal pairs

\text{cosec}\theta=\frac{1}{\sin\theta},\quad \sec\theta=\frac{1}{\cos\theta},\quad \cot\theta=\frac{1}{\tan\theta}

sin column

\sin0^\circ=0,\ \sin30^\circ=\frac12,\ \sin45^\circ=\frac{1}{\sqrt2},\ \sin60^\circ=\frac{\sqrt3}{2},\ \sin90^\circ=1

cos column

\cos\theta=\sin(90^\circ-\theta)

tan column

\tan0^\circ=0,\ \tan30^\circ=\frac{1}{\sqrt3},\ \tan45^\circ=1,\ \tan60^\circ=\sqrt3

Pythagorean identities

\sin^2\theta+\cos^2\theta=1,\quad 1+\tan^2\theta=\sec^2\theta,\quad 1+\cot^2\theta=\text{cosec}^2\theta

Fourth powers

\sin^4\theta+\cos^4\theta=1-2\sin^2\theta\cos^2\theta

Sixth powers

\sin^6\theta+\cos^6\theta=1-3\sin^2\theta\cos^2\theta

Product-to-sum bridge

\tan\theta=\frac{\sin\theta}{\cos\theta},\quad \cot\theta=\frac{\cos\theta}{\sin\theta}

Complementary pairs

\sin(90-\theta)=\cos\theta,\ \tan(90-\theta)=\cot\theta,\ \sec(90-\theta)=\text{cosec}\theta

Paired products

\tan\theta\,\tan(90^\circ-\theta)=1,\quad \sin\theta\,\text{cosec}\theta=1

Angle-equation rule

\sin(A+\theta)=\cos(B+\theta)\Rightarrow A+B+2\theta=90^\circ

From sin to the rest

\sin\theta=\frac{p}{h}\Rightarrow \cos\theta=\frac{\sqrt{h^2-p^2}}{h},\ \tan\theta=\frac{p}{\sqrt{h^2-p^2}}

Standard triplets

(3,4,5),\ (5,12,13),\ (7,24,25),\ (8,15,17),\ (9,40,41)

Sum-product of a ratio and reciprocal

x+\frac1x=k\Rightarrow x^2+\frac{1}{x^2}=k^2-2

Ranges

0\le\sin\theta\le1,\quad \sec\theta\ge1,\quad \sin\theta+\cos\theta\in[1,\sqrt2]

Weighted square form

a\sin^2\theta+b\cos^2\theta\in[\min(a,b),\max(a,b)]

AM-GM forms

a\tan^2\theta+b\cot^2\theta\ge2\sqrt{ab},\quad a\sin^2\theta+b\,\text{cosec}^2\theta\ge2\sqrt{ab}

Combined sec-cosec

\sec^2\theta+\text{cosec}^2\theta=2+\tan^2\theta+\cot^2\theta\ge4

Linear combination

a\sin\theta+b\cos\theta\text{ has maximum }\sqrt{a^2+b^2}