JEE MATH APEX
JEE MATH APEX
JEE MATH APEX
JEE MATH APEX
JEE MATH APEX
JEE MATH APEX
JEE MATH APEX
TRIGONOMETRY
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JEE MATH APEX
1. Basic Identities
📌 Fundamental Trigonometric Identities
• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = cosec²θ
• tan θ = sin θ / cos θ
• cot θ = cos θ / sin θ = 1/tan θ
• sec θ = 1/cos θ | cosec θ = 1/sin θ
• 1 + tan²θ = sec²θ
• 1 + cot²θ = cosec²θ
• tan θ = sin θ / cos θ
• cot θ = cos θ / sin θ = 1/tan θ
• sec θ = 1/cos θ | cosec θ = 1/sin θ
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2. Allied Angles
📌 Trigonometric Ratios of Allied Angles
• sin(−θ) = −sin θ | cos(−θ) = cos θ
• sin(90° − θ) = cos θ | cos(90° − θ) = sin θ
• sin(90° + θ) = cos θ | cos(90° + θ) = −sin θ
• sin(180° − θ) = sin θ | cos(180° − θ) = −cos θ
• sin(180° + θ) = −sin θ | cos(180° + θ) = −cos θ
• sin(270° − θ) = −cos θ | cos(270° − θ) = −sin θ
• sin(270° + θ) = −cos θ | cos(270° + θ) = sin θ
• sin(360° ± θ) = ±sin θ | cos(360° ± θ) = cos θ
• sin(90° − θ) = cos θ | cos(90° − θ) = sin θ
• sin(90° + θ) = cos θ | cos(90° + θ) = −sin θ
• sin(180° − θ) = sin θ | cos(180° − θ) = −cos θ
• sin(180° + θ) = −sin θ | cos(180° + θ) = −cos θ
• sin(270° − θ) = −cos θ | cos(270° − θ) = −sin θ
• sin(270° + θ) = −cos θ | cos(270° + θ) = sin θ
• sin(360° ± θ) = ±sin θ | cos(360° ± θ) = cos θ
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3. Compound Angle Formulas
📌 Sum and Difference Formulas
• sin(A + B) = sin A cos B + cos A sin B
• sin(A − B) = sin A cos B − cos A sin B
• cos(A + B) = cos A cos B − sin A sin B
• cos(A − B) = cos A cos B + sin A sin B
• tan(A + B) = (tan A + tan B) / (1 − tan A tan B)
• tan(A − B) = (tan A − tan B) / (1 + tan A tan B)
• cot(A + B) = (cot A cot B − 1) / (cot A + cot B)
• cot(A − B) = (cot A cot B + 1) / (cot B − cot A)
• sin(A − B) = sin A cos B − cos A sin B
• cos(A + B) = cos A cos B − sin A sin B
• cos(A − B) = cos A cos B + sin A sin B
• tan(A + B) = (tan A + tan B) / (1 − tan A tan B)
• tan(A − B) = (tan A − tan B) / (1 + tan A tan B)
• cot(A + B) = (cot A cot B − 1) / (cot A + cot B)
• cot(A − B) = (cot A cot B + 1) / (cot B − cot A)
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4. Double Angle Formulas
📌 Double Angle Formulas
• sin 2A = 2 sin A cos A = 2 tan A / (1 + tan²A)
• cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A
• cos 2A = (1 − tan²A) / (1 + tan²A)
• tan 2A = 2 tan A / (1 − tan²A)
• cot 2A = (cot²A − 1) / 2 cot A
• 1 + cos 2A = 2cos²A | 1 − cos 2A = 2sin²A
• cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A
• cos 2A = (1 − tan²A) / (1 + tan²A)
• tan 2A = 2 tan A / (1 − tan²A)
• cot 2A = (cot²A − 1) / 2 cot A
• 1 + cos 2A = 2cos²A | 1 − cos 2A = 2sin²A
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5. Triple Angle Formulas
📌 Triple Angle Formulas
• sin 3A = 3 sin A − 4 sin³A
• cos 3A = 4 cos³A − 3 cos A
• tan 3A = (3 tan A − tan³A) / (1 − 3 tan²A)
• sin³A = (3 sin A − sin 3A) / 4
• cos³A = (3 cos A + cos 3A) / 4
• cos 3A = 4 cos³A − 3 cos A
• tan 3A = (3 tan A − tan³A) / (1 − 3 tan²A)
• sin³A = (3 sin A − sin 3A) / 4
• cos³A = (3 cos A + cos 3A) / 4
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6. Half Angle Formulas
📌 Half Angle Formulas
• sin(A/2) = ±√[(1 − cos A)/2]
• cos(A/2) = ±√[(1 + cos A)/2]
• tan(A/2) = ±√[(1 − cos A)/(1 + cos A)]
• tan(A/2) = sin A / (1 + cos A) = (1 − cos A) / sin A
• 1 + cos A = 2cos²(A/2) | 1 − cos A = 2sin²(A/2)
• cos(A/2) = ±√[(1 + cos A)/2]
• tan(A/2) = ±√[(1 − cos A)/(1 + cos A)]
• tan(A/2) = sin A / (1 + cos A) = (1 − cos A) / sin A
• 1 + cos A = 2cos²(A/2) | 1 − cos A = 2sin²(A/2)
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7. Sum to Product Formulas
📌 Sum and Difference to Product
• sin C + sin D = 2 sin[(C+D)/2] cos[(C−D)/2]
• sin C − sin D = 2 cos[(C+D)/2] sin[(C−D)/2]
• cos C + cos D = 2 cos[(C+D)/2] cos[(C−D)/2]
• cos C − cos D = −2 sin[(C+D)/2] sin[(C−D)/2]
• cos C − cos D = 2 sin[(C+D)/2] sin[(D−C)/2]
• sin C − sin D = 2 cos[(C+D)/2] sin[(C−D)/2]
• cos C + cos D = 2 cos[(C+D)/2] cos[(C−D)/2]
• cos C − cos D = −2 sin[(C+D)/2] sin[(C−D)/2]
• cos C − cos D = 2 sin[(C+D)/2] sin[(D−C)/2]
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8. Product to Sum Formulas
📌 Product to Sum
• 2 sin A cos B = sin(A+B) + sin(A−B)
• 2 cos A sin B = sin(A+B) − sin(A−B)
• 2 cos A cos B = cos(A+B) + cos(A−B)
• 2 sin A sin B = cos(A−B) − cos(A+B)
• sin A sin B = [cos(A−B) − cos(A+B)]/2
• 2 cos A sin B = sin(A+B) − sin(A−B)
• 2 cos A cos B = cos(A+B) + cos(A−B)
• 2 sin A sin B = cos(A−B) − cos(A+B)
• sin A sin B = [cos(A−B) − cos(A+B)]/2
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9. General Solutions
📌 General Solutions of Trigonometric Equations
• sin θ = 0 ⇒ θ = nπ, n ∈ Z
• cos θ = 0 ⇒ θ = (2n+1)π/2, n ∈ Z
• tan θ = 0 ⇒ θ = nπ, n ∈ Z
• sin θ = sin α ⇒ θ = nπ + (−1)ⁿα, n ∈ Z
• cos θ = cos α ⇒ θ = 2nπ ± α, n ∈ Z
• tan θ = tan α ⇒ θ = nπ + α, n ∈ Z
• sin²θ = sin²α ⇒ θ = nπ ± α, n ∈ Z
• cos²θ = cos²α ⇒ θ = nπ ± α, n ∈ Z
• cos θ = 0 ⇒ θ = (2n+1)π/2, n ∈ Z
• tan θ = 0 ⇒ θ = nπ, n ∈ Z
• sin θ = sin α ⇒ θ = nπ + (−1)ⁿα, n ∈ Z
• cos θ = cos α ⇒ θ = 2nπ ± α, n ∈ Z
• tan θ = tan α ⇒ θ = nπ + α, n ∈ Z
• sin²θ = sin²α ⇒ θ = nπ ± α, n ∈ Z
• cos²θ = cos²α ⇒ θ = nπ ± α, n ∈ Z
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10. Inverse Trigonometric Functions
📌 Principal Values & Identities
• sin⁻¹x + cos⁻¹x = π/2
• tan⁻¹x + cot⁻¹x = π/2
• sec⁻¹x + cosec⁻¹x = π/2
• tan⁻¹x + tan⁻¹y = tan⁻¹[(x+y)/(1−xy)], if xy < 1
• tan⁻¹x − tan⁻¹y = tan⁻¹[(x−y)/(1+xy)], if xy > −1
• sin⁻¹x + sin⁻¹y = sin⁻¹[x√(1−y²) + y√(1−x²)]
• 2 tan⁻¹x = tan⁻¹[2x/(1−x²)] = sin⁻¹[2x/(1+x²)] = cos⁻¹[(1−x²)/(1+x²)]
• tan⁻¹x + cot⁻¹x = π/2
• sec⁻¹x + cosec⁻¹x = π/2
• tan⁻¹x + tan⁻¹y = tan⁻¹[(x+y)/(1−xy)], if xy < 1
• tan⁻¹x − tan⁻¹y = tan⁻¹[(x−y)/(1+xy)], if xy > −1
• sin⁻¹x + sin⁻¹y = sin⁻¹[x√(1−y²) + y√(1−x²)]
• 2 tan⁻¹x = tan⁻¹[2x/(1−x²)] = sin⁻¹[2x/(1+x²)] = cos⁻¹[(1−x²)/(1+x²)]
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11. Sine & Cosine Rule
📌 Properties of Triangles
• Sine Rule: a/sin A = b/sin B = c/sin C = 2R
• Cosine Rule: cos A = (b² + c² − a²) / 2bc
• cos B = (c² + a² − b²) / 2ca
• cos C = (a² + b² − c²) / 2ab
• Projection Formula: a = b cos C + c cos B
• Area of triangle: Δ = (1/2)bc sin A = (1/2)ca sin B = (1/2)ab sin C
• Cosine Rule: cos A = (b² + c² − a²) / 2bc
• cos B = (c² + a² − b²) / 2ca
• cos C = (a² + b² − c²) / 2ab
• Projection Formula: a = b cos C + c cos B
• Area of triangle: Δ = (1/2)bc sin A = (1/2)ca sin B = (1/2)ab sin C
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12. Series in Trigonometry
📌 Sum of Trigonometric Series
• sin α + sin(α+β) + sin(α+2β) + … + sin[α+(n−1)β]
= [sin(nβ/2) / sin(β/2)] · sin[α + (n−1)β/2]
• cos α + cos(α+β) + cos(α+2β) + … + cos[α+(n−1)β]
= [sin(nβ/2) / sin(β/2)] · cos[α + (n−1)β/2]
• cos θ · cos 2θ · cos 4θ · … · cos 2n−1θ = sin(2nθ) / (2n sin θ)
= [sin(nβ/2) / sin(β/2)] · sin[α + (n−1)β/2]
• cos α + cos(α+β) + cos(α+2β) + … + cos[α+(n−1)β]
= [sin(nβ/2) / sin(β/2)] · cos[α + (n−1)β/2]
• cos θ · cos 2θ · cos 4θ · … · cos 2n−1θ = sin(2nθ) / (2n sin θ)
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13. Important Results (Advanced)
📌 Maximum, Minimum & Conditional Identities
• Max of a sin θ + b cos θ = √(a² + b²)
• Min of a sin θ + b cos θ = −√(a² + b²)
• If A + B + C = π, then tan A + tan B + tan C = tan A · tan B · tan C
• sin A + sin B + sin C = 4 cos(A/2) cos(B/2) cos(C/2)
• cos A + cos B + cos C = 1 + 4 sin(A/2) sin(B/2) sin(C/2)
• sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C
• cos 2A + cos 2B + cos 2C = −1 − 4 cos A cos B cos C
• Min of a sin θ + b cos θ = −√(a² + b²)
• If A + B + C = π, then tan A + tan B + tan C = tan A · tan B · tan C
• sin A + sin B + sin C = 4 cos(A/2) cos(B/2) cos(C/2)
• cos A + cos B + cos C = 1 + 4 sin(A/2) sin(B/2) sin(C/2)
• sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C
• cos 2A + cos 2B + cos 2C = −1 − 4 cos A cos B cos C
✅ TRIGONOMETRY ✅
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— JEE MATH APEX —
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