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MathsMediumClass 11

Trigonometric Equations — General solutions

Trigonometric Functions & Equations

8

JEE Qs

8%

Hard

75

min

Master the conversion of complex trigonometric expressions into standard forms and meticulously handle domain restrictions and potential extraneous solutions.

🧮 Key Formulas

If sin x = sin y => x = nπ + (-1)^n y, where n ∈ Z
If cos x = cos y => x = 2nπ ± y, where n ∈ Z
If tan x = tan y => x = nπ + y, where n ∈ Z
If sin x = 0 => x = nπ, where n ∈ Z
If cos x = 0 => x = (2n+1)π/2, where n ∈ Z
If tan x = 0 => x = nπ, where n ∈ Z
If sin^2 x = sin^2 y => x = nπ ± y, where n ∈ Z
If cos^2 x = cos^2 y => x = nπ ± y, where n ∈ Z
If tan^2 x = tan^2 y => x = nπ ± y, where n ∈ Z

✅ Key Points for JEE

  • 1Always simplify the given trigonometric equation into one of the fundamental forms (sin x = sin y, cos x = cos y, tan x = tan y) using identities and algebraic manipulation.
  • 2Be mindful of the domain of trigonometric functions (e.g., tan x, sec x are undefined at (2n+1)π/2) and ensure solutions obtained are valid.
  • 3When squaring both sides of an equation or dividing by a trigonometric term, always check for extraneous solutions or lost solutions, respectively.
  • 4If multiple trigonometric terms are involved, try to express them in terms of a single trigonometric function or convert to a product/sum form that allows factorization.
  • 5The general solution for equations involving squares (sin^2 x = sin^2 y, etc.) is unified as x = nπ ± y, which is a powerful shortcut.

⚠️ Common Mistakes

  • ✕Incorrectly applying the general solution formulas (e.g., using the sine general solution for a cosine equation).
  • ✕Losing solutions by dividing both sides by a trigonometric expression that could be zero (e.g., dividing by sin x instead of factoring).
  • ✕Introducing extraneous solutions when squaring both sides of an equation without verifying them in the original equation.
  • ✕Ignoring the domain restrictions for functions like tan x, cot x, sec x, cosec x, leading to invalid solutions.
  • ✕Forgetting to include 'n ∈ Z' (n belongs to integers) as part of the general solution.

📝 Practice Questions

See all

Q11.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

2025·MCQHard

Q7. If ∑13r=1 { sin( 4 +(r−1) 6 ) sin( π4 + rπ6 ) } (1) 10 (2) 4 (3) 2 (4) 8

2025·MCQHard

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

2025·MCQMedium

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

2025·MCQEasy

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

2025·NumericalMedium

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

2025·MCQMedium

NCERT Chapters

  • Class 11 Mathematics Ch 3: Trigonometric Functions