Question
Find the general solution of: sin 2x = √3/2
Solution — Step by Step
This question tests whether you know the general solution formula for the sine function. Writing only specific angles like x = π/6 would be incomplete and would cost you marks.
Step 1: Identify the principal value.
We need sin 2x = √3/2.
From the standard table: sin 60° = sin(π/3) = √3/2.
So the principal value of the angle for which sin equals √3/2 is π/3.
Step 2: Write the general solution formula for sine.
If sin θ = sin α, the general solution is:
General Solution for sin θ = sin α
θ = nπ + (−1)ⁿ α, where n ∈ ℤ (all integers)
Here, θ = 2x and α = π/3.
Step 3: Set up the general equation.
2x = nπ + (−1)ⁿ (π/3), where n ∈ ℤ
Step 4: Solve for x.
x = nπ/2 + (−1)ⁿ (π/6), where n ∈ ℤ
Step 5: Verify with specific values of n.
Let n = 0: x = 0 + π/6 = π/6 Check: sin(2 × π/6) = sin(π/3) = √3/2 ✓
Let n = 1: x = π/2 − π/6 = 3π/6 − π/6 = 2π/6 = π/3 Check: sin(2 × π/3) = sin(2π/3) = sin(π − π/3) = sin(π/3) = √3/2 ✓
Let n = 2: x = π + π/6 = 7π/6 Check: sin(2 × 7π/6) = sin(7π/3) = sin(π/3) = √3/2 ✓
General Solution
x = nπ/2 + (−1)ⁿ (π/6), n ∈ ℤ
Why This Works
Sine is a periodic function with period 2π. Within each period, it takes every value between −1 and 1 exactly twice (once in the first half-period and once in the second). The general solution formula captures all these occurrences across all periods.
The (−1)ⁿ factor is clever: when n is even, it adds α; when n is odd, it subtracts α. This alternation corresponds to the two solutions within each period — one at α and one at π − α (both give the same sine value because sin is symmetric about π/2).
Alternative Method: Checking if sin θ = sin(π − θ)
sin 2x = √3/2 = sin(π/3) or sin(π − π/3) = sin(2π/3)
So either: 2x = π/3 + 2nπ → x = π/6 + nπ
Or: 2x = 2π/3 + 2nπ → x = π/3 + nπ
Both families combined can be written as the single general solution above.
🎯 Exam Insider
In JEE Main, general solution questions always require the answer in terms of n ∈ ℤ (or n ∈ Z). Writing "x = π/6 and x = π/3" without the general form scores partial marks at best. Always include the "where n ∈ ℤ" part explicitly.
Common Mistake
⚠️ Common Mistake
Mistake: Using the cosine general solution formula for a sine equation.
For sin θ = sin α: θ = nπ + (−1)ⁿα
For cos θ = cos α: θ = 2nπ ± α
For tan θ = tan α: θ = nπ + α
These three are different formulas with different structures. Using the wrong formula — especially mixing up the sin and cos forms — is a very common JEE error. Write all three on a reference sheet and don't confuse them.