Solving cos(x/2) = -1/2

Trigonometry · Axiom Academy

EXAMPLE Solving Half-Angle Cosine Equations Learn to solve trigonometric equations with half-angles and understand period extension. Excellent work! You've successfully solved a half-angle cosine equation. Here's what we learned: Substitution Strategy: When you see cos(x/2), let u = x/2 to transform it into the simpler equation cos(u) = value. Reference Angles: cos(θ) = -1/2 occurs at θ = 2π/3 and θ = 4π/3 (in Quadrants II and III where cosine is negative). General Solutions: Add 2πn to capture all coterminal angles in the basic form before substituting back. Period Extension: The half-angle doubles the period! While cos(x) has period 2π, cos(x/2) has period 4π . This means multiplying your general solution by 2 gives you 4πn. Back-Substitution: Always remember to substitute back (u → x/2) and solve for the original variable by multiplying both sides by 2. This technique works for any half-angle equation. The key insight: dividing the argument by 2 multiplies the period by 2. Similarly, cos(x/3) would have period 6π!

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