Multiple Angles — sin 2A, cos 2A, tan 2A
Trigonometric Functions & Equations
8
JEE Qs
8%
Hard
60
min
Master all forms of cos 2A and their derivations; quick recall and manipulative skill with these identities are crucial for solving complex trigonometric problems efficiently.
🧮 Key Formulas
✅ Key Points for JEE
- 1All multiple angle formulas are derived from compound angle formulas by setting B=A (e.g., sin 2A = sin(A+A)).
- 2The cos 2A formulas have four forms; choose the most appropriate one based on the terms present in the problem or the desired form of the answer.
- 3The expressions (1 - cos 2A) = 2 sin^2 A and (1 + cos 2A) = 2 cos^2 A are extremely useful for simplification and are often encountered.
- 4Remember the 'tan A' forms for sin 2A, cos 2A, and tan 2A; they are crucial for expressing trigonometric expressions entirely in terms of tan A, especially useful in coordinate geometry and calculus.
⚠️ Common Mistakes
- ✕Confusing the denominator of sin 2A (in terms of tan A) which is (1 + tan^2 A) with that of tan 2A which is (1 - tan^2 A).
- ✕Incorrectly recalling or deriving the various forms of cos 2A, leading to errors in simplification.
- ✕Failing to recognize when to apply the 'reverse' forms like 2 sin^2 A = 1 - cos 2A or 2 cos^2 A = 1 + cos 2A quickly.
📝 Practice Questions
See allQ11.Let the range of the function f(x) = 6 + 16 cos x ⋅cos ( π3 −x) ⋅cos ( π3 + x) ⋅sin 3x ⋅cos 6x, x ∈R be [α, β] . Then the distance of the point (α, β) from the line 3x + 4y + 12 = 0 is : (1) 11 (2) 8 (3) 10 (4) 9 sin y > 0 and x(1) = π2 . Then
Q7. If ∑13r=1 { sin( 4 +(r−1) 6 ) sin( π4 + rπ6 ) } (1) 10 (2) 4 (3) 2 (4) 8
Q18.The sum of all values of θ ∈[0, 2π] satisfying 2 sin2 θ = cos 2θ and 2 cos2 θ = 3 sin θ is 2025 (22 Jan Shift 2) JEE Main Previous Year Paper (1) 4π (2) 5π6 (3) π (4) π 2
Q18.The value of (sin 70∘) (cot 10∘cot 70∘−1) is (1) 2/3 (2) 1 (3) 0 (4) 3/2 dx 1 1 1 , then 3( b + c) is equal to
Q22.If for some α, β; α ≤β, α + β −8 and sec2 (tan−1 α) + cosec2 (cot−1 β) −36, then α2 + β is_______. Q23. ⎡x⎤ Let A be a 3 × 3 matrix such that X TAX = O for all nonzero 3 × 1 matrices X = y . If ⎣z ⎦ ⎡ 1 ⎤ ⎡ 1 ⎤ ⎡1 ⎤ ⎡ 0 ⎤ A 1 = 4 , A 2 = 4 , and det(adj(2(A + 1))) −2α3β5γ, α, β, γ ∈N , then α2 + β2 + γ 2 ⎣ 1⎦ ⎣ −5 ⎦ ⎣1⎦ ⎣−8 ⎦ is_____. x ≥0. Then
Q20.If sin x + sin2 x = 1, x ∈(0, π2 ), then (cos12 x + tan12 x) + 3 (cos10 x + tan10 x + cos8 x + tan8 x) + (cos6 x + tan6 x) is equal to : (1) 4 (2) 1 (3) 3 (4) 2 π
NCERT Chapters
- Class 11 Mathematics Ch 3: Trigonometric Functions