11 mark for review the function f is given by f(θ)=cosθ. if all of the other necessary conditions are met…

Explanation:

Step1: Recall cosine - function properties

For the function $y = \cos\theta$, when $\theta=0$, $y = \cos(0)=1$. The general form of a cosine - function is $y = A\cos(B\theta - C)+D$. The period $T$ of the function $y=\cos\theta$ (where $B = 1$) is given by $T=\frac{2\pi}{|B|}=2\pi$, and the frequency $f=\frac{1}{T}=\frac{1}{2\pi}$.

Step2: Analyze each option

• Option A: The period of $y = \cos\theta$ is $2\pi$, not $\frac{1}{2\pi}$, and $\cos(0)=1\neq0$.
• Option B: The frequency of $y = \cos\theta$ is $\frac{1}{2\pi}$, but $\cos(0)=1\neq0$.
• Option C: The period of $y = \cos\theta$ is $2\pi$, not $\frac{1}{2\pi}$.
• Option D: The frequency of the function $y = \cos\theta$ is $f=\frac{1}{2\pi}$ (since $T = 2\pi$ and $f=\frac{1}{T}$), and when $\theta = 0$, $f(0)=\cos(0)=1$.

Answer:

D. A scenario where the frequency is $\frac{1}{2\pi}$ and the initial value for $\theta = 0$ is 1