Tangent Half-Angle
Expresses the tangent of a half angle.
Formula
Variables
Example
Used in integration substitutions
How it works
This identity rewrites the tangent of a half-angle in terms of the full angle’s sine and cosine, which is useful because it replaces a smaller-angle tangent with quantities that may be easier to compute or simplify. It comes from standard half-angle identities: starting with 1- cos\theta = 2\sin^2(\theta/2) and sin\theta = 2\sin(\theta/2)\cos(\theta/2), dividing the two gives tan(\theta/2). The formula is especially powerful when trigonometric expressions need to be converted into algebraic forms or when solving equations involving half-angles.
When to use it
Use this identity when you know \sin\theta and \cos\theta but need \tan(\theta/2), or when simplifying expressions that contain half-angles. It also appears in advanced trigonometric substitutions, integration, and geometric derivations where converting between full-angle and half-angle quantities makes the problem easier.
Worked example, step by step
Find \tan(\theta/2) when \theta = 60^\circ.\nStep 1: Substitute into the formula: \tan(\theta/2) = (1 - \cos\theta)/\sin\theta\nStep 2: Use exact values: \cos 60^\circ = 1/2 and \sin 60^\circ = \sqrt{3}/2\nStep 3: Compute the numerator: 1 - 1/2 = 1/2\nStep 4: Divide: (1/2)/(\sqrt{3}/2) = 1/\sqrt{3}\nStep 5: Rationalize if desired: 1/\sqrt{3} = \sqrt{3}/3\nFinal answer: \tan(30^\circ) = \sqrt{3}/3
Common mistakes to avoid
- Using degrees in the calculator but plugging in radian values, or vice versa
- Forgetting that the denominator is \sin\theta, not \cos\theta
- Mixing up \tan(\theta/2) with \tan\theta/2, which are not the same expression
Did You Know?
The tangent half-angle substitution can turn any trigonometric integral into an ordinary algebraic one.
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