Trigonometric Equations — General Solutions
Trigonometric Functions & Equations
40
JEE Qs
12%
Hard
75
min
Master the standard general solution formulas and prioritize factorization and identity application to reduce complex equations into solvable forms, always checking for domain validity.
🧮 Key Formulas
✅ Key Points for JEE
- 1Always transform given trigonometric equations into one of the standard forms (sin X = sin Y, cos X = cos Y, tan X = tan Y, or their squared versions) before applying general solution formulas.
- 2Factorization is crucial: never cancel common trigonometric terms if they can be zero; instead, take them common to avoid losing valid solutions.
- 3Pay close attention to the domain of the trigonometric functions (e.g., tan x is undefined at (2n+1)π/2) and ensure the obtained solutions are valid within these restrictions.
- 4When an equation involves different trigonometric ratios or angles, use appropriate identities to convert them into a single ratio or angle, or to facilitate factorization.
⚠️ Common Mistakes
- ✕Cancelling common trigonometric factors from both sides of an equation, which leads to losing solutions where that factor is zero.
- ✕Incorrectly applying the general solution formulas, particularly the `(-1)^n` term for sine equations or confusing `nπ` with `2nπ`.
- ✕Ignoring the implicit domain restrictions of functions like tan x, cot x, sec x, and cosec x, resulting in extraneous solutions.
- ✕Not checking for extraneous solutions after squaring both sides of an equation or when converting to a new form.
📝 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 Maths Ch 3: Trigonometric Functions