UGA Math Placement Practice Exam

The formula for \(\tan\left(\frac{x}{2}\right)\) is derived from the half-angle identities in trigonometry. One of the most useful forms of the half-angle identity for tangent is expressed as: \[ \tan\left(\frac{x}{2}\right) = \frac{\sin(x)}{1 + \cos(x)} \] and also can be expressed as: \[ \tan\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 - \cos(x)}{1 + \cos(x)}} \] The selected response aligns with the latter form, which is useful especially when calculating the tangent of half angles in certain applications. The derivation of this formula involves fundamental trigonometric identities where, by manipulating the sine and cosine functions through double angle formulas, this expression emerges. The square root notation indicates that the sign of \(\tan\left(\frac{x}{2}\right)\) depends on the quadrant in which the angle \(\frac{x}{2}\) lies. Thus, the correct answer provides an alternate representation of \(\tan\left(\frac{x}{2}\right)\) that is valid for every angle \(x\