Half Angle: Tangent

This identity rewrites the tangent of half an angle using the sine and cosine of the full angle, so it is useful when \(\theta\) is known but \(\theta/2\) is not. It comes from the standard half-angle identities: \(1-\cos\theta = 2\sin^2(\theta/2)\) and \(\sin\theta = 2\sin(\theta/2)\cos(\theta/2)\); dividing them gives \(\tan(\theta/2)\). It is especially helpful because it avoids directly evaluating \(\theta/2\) and can convert tricky trigonometric expressions into algebraic ones.

Use this when you need \(\tan(\theta/2)\) but only know \(\sin\theta\) and \(\cos\theta\), or when simplifying expressions involving half-angles in trig equations and integrals. It is also useful in geometry and physics problems where full-angle measurements are given but a half-angle ratio is required.

• Using the formula with \(\theta\) already halved, instead of substituting the full angle into \(\cos\theta\) and \(\sin\theta\)
• Forgetting that \(\sin\theta\) must be nonzero, since the fraction is undefined when \(\sin\theta = 0\)
• Mixing up the related half-angle forms, such as using \(\frac{1-\cos\theta}{\sin\theta}\) when the problem actually needs a signed version based on the quadrant of \(\theta/2\)