Solving sin(3x) = √3/2

Trigonometry · Axiom Academy

EXAMPLE Solving sin(3x) = √3/2 Master the substitution method for multiple-angle trigonometric equations and predict the number of solutions in a given interval. Excellent work! You've mastered solving multiple-angle trigonometric equations. Here's what we learned: Substitution Strategy: Let u = 3x to transform the equation into a simpler form. This turns a complex multiple-angle problem into a basic trig equation. Expand the Range: If x ∈ [0, 2π), then 3x ∈ [0, 6π). This means we search for solutions across 3 complete periods of the sine function. Count Solutions per Period: For sin(u) = √3/2, there are 2 solutions per period (π/3 and 2π/3). With 3 periods, we get 2 × 3 = 6 total solutions. Convert Back: After finding all u values, divide by 3 to get the corresponding x values: x = u/3. General Pattern: For sin(nx) = k, the number of solutions in [0, 2π) is typically 2n (assuming |k| ≤ 1). This substitution method is essential for solving any multiple-angle trigonometric equation. You can apply the same technique to cosine, tangent, and other trig functions!

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