Question

18 mark for review
the functions $f$ and $g$ are given by $f(\theta)=\cos\theta$ and $g(\theta)=\sin\theta$. on which of the following intervals are both $f$ and $g$ increasing?
a $0<\theta<\frac{\pi}{2}$
b $\frac{\pi}{2}<\theta<\pi$
c $\pi<\theta<\frac{3\pi}{2}$
d $\frac{3\pi}{2}<\theta<2\pi$

Explanation:

Step1: Find derivative of $f(\theta)$

$f'(\theta) = \frac{d}{d\theta}\cos\theta = -\sin\theta$

Step2: Find derivative of $g(\theta)$

$g'(\theta) = \frac{d}{d\theta}\sin\theta = \cos\theta$

Step3: Analyze interval A ($0<\theta<\frac{\pi}{2}$)

$f'(\theta)=-\sin\theta <0$ ($f$ decreasing), $g'(\theta)=\cos\theta>0$ ($g$ increasing)

Step4: Analyze interval B ($\frac{\pi}{2}<\theta<\pi$)

$f'(\theta)=-\sin\theta <0$ ($f$ decreasing), $g'(\theta)=\cos\theta<0$ ($g$ decreasing)

Step5: Analyze interval C ($\pi<\theta<\frac{3\pi}{2}$)

$f'(\theta)=-\sin\theta >0$ ($f$ increasing), $g'(\theta)=\cos\theta<0$ ($g$ decreasing)

Step6: Analyze interval D ($\frac{3\pi}{2}<\theta<2\pi$)

$f'(\theta)=-\sin\theta >0$ ($f$ increasing), $g'(\theta)=\cos\theta>0$ ($g$ increasing)

Answer:

D. $\frac{3\pi}{2} < \theta < 2\pi$