Ohio Assessments for Educators (OAE) Mathematics Practice Exam

Which formula represents cos(θ/2) in terms of cos(θ)?

1. ±√((1+cosθ)/2)

2. ±√((1-cosθ)/2)

3. √(1-cos²θ)

4. 1-2sin²(θ/2)

The correct answer is: ±√((1+cosθ)/2)

The formula for cos(θ/2) in terms of cos(θ) is derived from the half-angle identity for cosine. Specifically, the half-angle identity states that: \[ cos(θ/2) = ±√((1 + cos(θ))/2) \] This relation is established by using trigonometric identities and the unit circle, where the cosine of an angle is associated with the x-coordinate of a point on the circle. When halving an angle, the new angle (θ/2) will correspond to a point that is still on the unit circle, and thus its cosine can be expressed in terms of the cosine of the original angle (θ) via the formula above. The positive and negative signs account for the fact that cosine can be either positive or negative depending on the quadrant in which the angle θ/2 lies. This identity allows us to compute cosine values for half-angles easily, which can be incredibly useful in solving various trigonometric problems or simplifying expressions. In contrast, the other provided formulas do not accurately represent cos(θ/2) in terms of cos(θ). For example, the option that utilizes (1-cosθ) does not align with the half-angle formula for cosine