Ohio Assessments for Educators (OAE) Mathematics Practice Exam

What is the value of sin(θ/2) expressed in terms of cos(θ)?

1. ±√((1-cosθ)/2)

2. ±√((1+cosθ)/2)

3. ±√((cosθ-1)/2)

4. sinθ/2

The correct answer is: ±√((1-cosθ)/2)

The value of sin(θ/2) can be expressed in terms of cos(θ) using the half-angle identity for sine. The half-angle identity states that: \[ \sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos\theta}{2}} \] This formula is derived from the cosine double angle identity, which relates angle cosines to angle sines. Specifically, we take the identity for sine in terms of cosine and apply it to half-angles. The expression \(\sqrt{\frac{1 - \cos\theta}{2}}\) indicates that the sine of half the angle is positive or negative, hence the ± symbol. This takes into account that the sine function can be both positive and negative depending on the quadrant in which the angle \(\frac{\theta}{2}\) lies. The other options presented do not correctly represent the relationship between sin(θ/2) and cos(θ). For example, the choice that includes \(\sinθ/2\) does not align with the half-angle identities. The formula \(\sqrt{\frac{1 + \cos\theta}{2}}\) corresponds to the cosine half-angle identity rather than sine