Trigonometric Ratios & Identities

Trigonometry: Basics

• Trigonometry deals with measurement of triangles.
• Angle measurement systems:
  • Sexagesimal (Degree): 1° = 60′, 1′ = 60″
  • Centesimal (Grade): 1 right angle = 100g
  • Circular (Radian): angle subtended by arc = radius
• Relation: 90° = 100g = π/2 radians

Trigonometric Ratios (Right-angled Triangle)

• sin θ = Perpendicular / Hypotenuse
• cos θ = Base / Hypotenuse
• tan θ = Perpendicular / Base
• cosec θ = Hypotenuse / Perpendicular
• sec θ = Hypotenuse / Base
• cot θ = Base / Perpendicular

Signs of Trigonometric Ratios

• I Quadrant: All ratios positive
• II Quadrant: sin, cosec positive
• III Quadrant: tan, cot positive
• IV Quadrant: cos, sec positive

Domain & Range (Important)

• sin x, cos x: Domain = R, Range = [−1, 1]
• tan x: Domain = R − {(2n+1)π/2}, Range = R
• cot x: Domain = R − {nπ}, Range = R
• sec x, cosec x: Range = R − (−1, 1)

Standard Angle Values

• sin 0° = 0, sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2, sin 90° = 1
• cos 0° = 1, cos 30° = √3/2, cos 45° = 1/√2, cos 60° = 1/2, cos 90° = 0
• tan 0° = 0, tan 45° = 1, tan 60° = √3

Sum & Difference Formulae

• sin(A ± B) = sinA cosB ± cosA sinB
• cos(A ± B) = cosA cosB ∓ sinA sinB
• tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB)
• sin2A = 2 sinA cosA
• cos2A = 1 − 2sin²A = 2cos²A − 1
• tan2A = 2tanA / (1 − tan²A)

Transformation Formulae

Sum → Product

• sinC + sinD = 2 sin((C+D)/2) cos((C−D)/2)
• sinC − sinD = 2 cos((C+D)/2) sin((C−D)/2)
• cosC + cosD = 2 cos((C+D)/2) cos((C−D)/2)
• cosC − cosD = −2 sin((C+D)/2) sin((C−D)/2)

Product → Sum

• 2 sinA cosB = sin(A+B) + sin(A−B)
• 2 cosA cosB = cos(A+B) + cos(A−B)
• 2 sinA sinB = cos(A−B) − cos(A+B)

Important Angle Results

• sin 15° = (√3 − 1)/(2√2)
• cos 15° = (√3 + 1)/(2√2)
• tan 15° = 2 − √3
• sin 18° = (√5 − 1)/4
• cos 36° = (√5 + 1)/4

Formulas for Three Angles

• sin(A+B+C) = sinA cosB cosC + cosA sinB cosC + cosA cosB sinC − sinA sinB sinC
• cos(A+B+C) = cosA cosB cosC − sinA sinB cosC − sinA cosB sinC − cosA sinB sinC
• tan(A+B+C) = (tanA + tanB + tanC − tanA tanB tanC) / (1 − tanA tanB − tanB tanC − tanC tanA)

Conditional Identities (A + B + C = π)

• sin2A + sin2B + sin2C = 2 + 2 cosA cosB cosC
• cos2A + cos2B + cos2C = 1 − 2 cosA cosB cosC
• tanA + tanB + tanC = tanA tanB tanC

Componendo & Dividendo

• If p/q = a/b, then (p+q)/(p−q) = (a+b)/(a−b)
• Also (p−q)/(p+q) = (a−b)/(a+b)

Useful Results (JEE Favourite)

• a sinx + b cosx ∈ [−√(a²+b²), √(a²+b²)]
• sin²x + cosec²x ≥ 2
• cos²x + sec²x ≥ 2
• tan²x + cot²x ≥ 2

Trigonometric Series

• sinα + sin(α+β) + … (n terms) = sin(nβ/2) · sin(α+(n−1)β/2) / sin(β/2)
• cosα + cos(α+β) + … (n terms) = sin(nβ/2) · cos(α+(n−1)β/2) / sin(β/2)