10 mark for review the functions f and g are given by f(θ) = sin θ and g(θ) = sin(θ + π/2). on the interval…

10 mark for review the functions f and g are given by f(θ) = sin θ and g(θ) = sin(θ + π/2). on the interval 0 ≤ θ ≤ 2π, how many solutions exist to f(θ) = g(θ)? a zero b one c two d infinitely many

10 mark for review the functions f and g are given by f(θ) = sin θ and g(θ) = sin(θ + π/2). on the interval 0 ≤ θ ≤ 2π, how many solutions exist to f(θ) = g(θ)? a zero b one c two d infinitely many

Answer

Explanation:

Step1: Set the two - functions equal

Set $\sin\theta=\sin(\theta +\frac{\pi}{2})$. Using the trigonometric identity $\sin(A + B)=\sin A\cos B+\cos A\sin B$, we have $\sin(\theta+\frac{\pi}{2})=\sin\theta\cos\frac{\pi}{2}+\cos\theta\sin\frac{\pi}{2}=\cos\theta$. So the equation becomes $\sin\theta=\cos\theta$.

Step2: Solve the equation

If $\sin\theta=\cos\theta$, then $\frac{\sin\theta}{\cos\theta}=1$ (assuming $\cos\theta\neq0$), i.e., $\tan\theta = 1$. The solutions of $\tan\theta=1$ in the interval $0\leq\theta\leq2\pi$ are $\theta=\frac{\pi}{4}$ and $\theta=\frac{5\pi}{4}$.

Answer:

C. Two