10 mark for review the functions f and g are given by f(θ)=cscθ and g(θ)=sinθ. for what values of θ on the…

10 mark for review the functions f and g are given by f(θ)=cscθ and g(θ)=sinθ. for what values of θ on the interval 0≤θ≤2π does f(θ)=g(θ)? a π/2 only b π/4 and 5π/4 c π/2 and 3π/2 d 0,π, and 2π

10 mark for review the functions f and g are given by f(θ)=cscθ and g(θ)=sinθ. for what values of θ on the interval 0≤θ≤2π does f(θ)=g(θ)? a π/2 only b π/4 and 5π/4 c π/2 and 3π/2 d 0,π, and 2π

Answer

Explanation:

Step1: Recall the definition of cosecant

Since $\csc\theta=\frac{1}{\sin\theta}$, the equation $f(\theta) = g(\theta)$ becomes $\frac{1}{\sin\theta}=\sin\theta$.

Step2: Cross - multiply

Cross - multiplying gives $1=\sin^{2}\theta$.

Step3: Solve for $\sin\theta$

Taking the square root of both sides, we have $\sin\theta=\pm1$.

Step4: Find $\theta$ values in the given interval

When $\sin\theta = 1$, $\theta=\frac{\pi}{2}$; when $\sin\theta=- 1$, $\theta=\frac{3\pi}{2}$ in the interval $0\leq\theta\leq2\pi$.

Answer:

C. $\frac{\pi}{2}$ and $\frac{3\pi}{2}$