Half Angle: Cosine

This identity rewrites the cosine of half an angle entirely in terms of the cosine of the full angle, so it is useful when you know cos(θ) but need cos(θ/2). It comes from the double-angle formula cos(θ) = 2cos²(θ/2) - 1; solving that equation for cos(θ/2) gives the square root form, with the ± sign reflecting that a half-angle can lie in different quadrants. The correct sign must be chosen from the angle’s quadrant or from additional information about the situation.

Use this when simplifying trigonometric expressions that contain half-angles, or when an equation gives cos(θ) and you need cos(θ/2) without directly finding θ. It is also common in calculus, geometry, and physics problems involving symmetry, rotations, or identities.

• Forgetting the ± sign and choosing a sign without checking the quadrant
• Substituting θ/2 into the formula instead of using θ as the full angle
• Dropping the outer square root or simplifying the fraction inside it incorrectly