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Half-Angle Formulas

Definition

These formulas express trigonometric functions of θ2\frac{\theta}{2} in terms of the original angle θ\theta.

  • sin⁡(θ2)=±1−cos⁡θ2\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos \theta}{2}}
  • cos⁡(θ2)=±1+cos⁡θ2\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos \theta}{2}}
  • tan⁡(θ2)=±1−cos⁡θ1+cos⁡θ=1−cos⁡θsin⁡θ=sin⁡θ1+cos⁡θ\tan\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} = \frac{1 - \cos \theta}{\sin \theta} = \frac{\sin \theta}{1 + \cos \theta}

Why It Matters

Complex waveforms and signals often involve “sub-harmonics.” Half-angle formulas matter because they are the “algebraic gears” that allow us to shift between different frequencies. They allow us to find exact values of trigonometric functions for smaller angles and are essential for various algebraic substitutions in calculus.

Core Concepts

  • Sine Half-Angle

    • sin⁡(θ2)=±1−cos⁡θ2\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos \theta}{2}}
      • How to read: “The sine of theta divided by two equals plus or minus the square root of the quantity one minus the cosine of theta, all over two.”
      • Meaning / when to use: Find exact values of half-angles; choose ±\pm by the quadrant of θ/2\theta/2.

  • Cosine Half-Angle

    • cos⁡(θ2)=±1+cos⁡θ2\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos \theta}{2}}
      • How to read: “The cosine of theta divided by two equals plus or minus the square root of the quantity one plus the cosine of theta, all over two.”
      • Meaning / when to use: Companion half-angle form; choose ±\pm by the quadrant of θ/2\theta/2.

  • Tangent Half-Angle

    • tan⁡(θ2)=±1−cos⁡θ1+cos⁡θ=1−cos⁡θsin⁡θ=sin⁡θ1+cos⁡θ\tan\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} = \frac{1 - \cos \theta}{\sin \theta} = \frac{\sin \theta}{1 + \cos \theta}
      • How to read: “The tangent of theta divided by two equals plus or minus the square root of the ratio of the quantity one minus the cosine of theta to the quantity one plus the cosine of theta, or the ratio of the quantity one minus the cosine of theta to the sine of theta, or the ratio of the sine of theta to the quantity one plus the cosine of theta.”
      • Meaning / when to use: Rationalized forms avoid nested radicals; common in Weierstrass substitutions and angle-bisection problems.

  • Sign rule for half-angles: The ± is chosen according to the quadrant in which the half-angle itself lies, not the original angle. sin(θ/2) is positive in quadrants I and II, negative in III and IV, etc.

  • Derivation note: Half-angle formulas are obtained by solving the double-angle cosine identity for sinθ or cosθ and then taking square roots (hence the ±).

Connected Concepts

Connected notes