Ohio Assessments for Educators (OAE) Mathematics Practice Exam

How is tan(θ/2) defined in relation to sin(θ) and cos(θ)?

1. sinθ/(1+cosθ)

2. sinθ/(1-cosθ)

3. cosθ/sinθ

4. 1/cos(θ/2)

The correct answer is: sinθ/(1+cosθ)

The tangent of half an angle, tan(θ/2), can be expressed in terms of the sine and cosine of the angle θ. The correct definition is derived from trigonometric identities. Specifically, tan(θ/2) is related to the sine and cosine of θ through the identity that states: \[ \tan(\frac{\theta}{2}) = \frac{\sin(\theta)}{1 + \cos(\theta)} \] This relationship is useful in various mathematical contexts, particularly in integration and simplification of expressions involving trigonometric functions. The formula effectively simplifies the computation of the tangent for half angles. This formulation derives from using the half-angle identities and the fundamental definitions of sine and cosine in terms of the unit circle. When you break down the components of sine and cosine, the transformation reflects how half an angle alters the relationships between these fundamental trigonometric ratios. Thus, the choice indicating that tan(θ/2) is equal to sin(θ)/(1 + cos(θ)) is accurate, showcasing how it leverages understanding of sine and cosine to yield a simpler measure of the tangent at half the angle.