all trigo formulas in one

JEEsociety - Trigonometry 1


JEEsociety
Chapter: Trigonometric Ratios & Identities


1 Fundamental Concepts

Basic Trigonometric Identities

• sin2 θ + cos2 θ = 1

• sec2 θ − tan2 θ = 1

• csc2 θ − cot2 θ = 1

• If sec θ + tan θ = k, then sec θ − tan θ = 1/k.

• If csc θ + cot θ = k, then csc θ − cot θ = 1/k.



1.1 Signs of T-Ratios in Quadrants (ASTC Rule)

90◦

S A
Quadrant II Quadrant I
sin, csc Positive All Positive

180◦ 0◦ , 360◦


Quadrant III Quadrant IV
tan, cot Positive cos, sec Positive
T C

270◦

Figure 1: The ASTC rule for signs of trigonometric functions.

, JEEsociety - Trigonometry 2


2 Trigonometric Formulas

2.1 Allied Angles

Common Allied Angle Transformations

sin(−θ) = − sin θ sin(90◦ − θ) = cos θ sin(180◦ − θ) = sin θ sin(270◦ − θ) = − cos θ
cos(−θ) = cos θ cos(90◦ − θ) = sin θ cos(180◦ − θ) = − cos θ cos(270◦ − θ) = − sin θ
tan(−θ) = − tan θ sin(90◦ + θ) = cos θ sin(180◦ + θ) = − sin θ sin(270◦ + θ) = − cos θ
cos(90◦ + θ) = − sin θ cos(180◦ + θ) = − cos θ cos(270◦ + θ) = sin θ



2.2 Sum and Difference Formulas

Key Formula

• sin(A ± B) = sin A cos B ± cos A sin B

• cos(A ± B) = cos A cos B ∓ sin A sin B
tan A±tan B
• tan(A ± B) = 1∓tan A tan B

cot A cot B∓1
• cot(A ± B) = cot B±cot A

• sin(A + B + C) = sin A cos B cos C + · · · − sin A sin B sin C

• cos(A + B + C) = cos A cos B cos C − · · · − cos A sin B sin C
tan A+tan B+tan C−tan A tan B tan C
• tan(A + B + C) = 1−tan A tan B−tan B tan C−tan C tan A