Ohio Assessments for Educators (OAE) Mathematics Practice Exam

What is the expression for cot(θ/2)?

1. sinθ/(1-cosθ)

2. sinθ/(1+cosθ)

3. (cotθ-tanθ)/2

4. sin(θ/2)/cos(θ/2)

The correct answer is: (cotθ-tanθ)/2

The expression for cot(θ/2) can be derived using trigonometric identities. Cotangent is defined as the ratio of the cosine of an angle to the sine of the same angle. For cot(θ/2), this means: \[ \cot\left(\frac{\theta}{2}\right) = \frac{\cos\left(\frac{\theta}{2}\right)}{\sin\left(\frac{\theta}{2}\right)}. \] There is an alternative identity that can be utilized to express cot(θ/2) using cotangent and tangent of the entire angle θ. The half-angle formulas for tangent and cotangent allow us to express cot(θ/2) in terms of those functions for θ. Specifically, one can relate cot(θ/2) to cot(θ) and tan(θ) by the identity: \[ \cot\left(\frac{\theta}{2}\right) = \frac{\cot(\theta) - \tan(\theta)}{2}. \] This relationship illustrates how cotangent values for the full angle can be transformed into an equivalent relationship involving half the angle. Thus, this formulation is mathematically valid and is why this choice accurately represents cot(θ/