For 0 ≤ x ≤ 2π, sin x and cos x are both decreasing functions in the interval ________.

Understanding Decreasing Functions for sin x and cos x

To find the interval where both $\sin x$ and $\cos x$ are decreasing functions, we need to analyze their behavior in the given range $0\le x\le 2\pi$. A function is decreasing when its derivative is negative.

Derivatives of sin x and cos x

First, let's find the derivatives of the two functions:

- The derivative of $\sin x$ is $\frac{d}{dx}(\sin x)=\cos x$.
- The derivative of $\cos x$ is $\frac{d}{dx}(\cos x)=-\sin x$.

Intervals Where sin x is Decreasing

The function $\sin x$ is decreasing when its derivative, $\cos x$, is negative.

In the interval $0\le x\le 2\pi$, $\cos x$ is negative in the second and third quadrants.

- Quadrant II: $\frac{\pi }{2}<x<\pi$
- Quadrant III: $\pi <x<\frac{3\pi }{2}$

Therefore, $\sin x$ is decreasing in the interval $\left(\frac{\pi }{2},\frac{3\pi }{2}\right)$.

Intervals Where cos x is Decreasing

The function $\cos x$ is decreasing when its derivative, $-\sin x$, is negative, which means $\sin x$ must be positive.

Alternatively, we can directly look at the behavior of $\cos x$. $\cos x$ starts at 1 (at $x=0$), decreases to 0 (at $x=\frac{\pi }{2}$), decreases further to -1 (at $x=\pi$), increases to 0 (at $x=\frac{3\pi }{2}$), and increases back to 1 (at $x=2\pi$).

So, $\cos x$ is decreasing in the first and second quadrants.

- Quadrant I: $0<x<\frac{\pi }{2}$
- Quadrant II: $\frac{\pi }{2}<x<\pi$

Therefore, $\cos x$ is decreasing in the interval $\left(0,\pi \right)$.

Finding the Common Interval

We need the interval where both $\sin x$ and $\cos x$ are decreasing. We compare the intervals found above:

- $\sin x$ decreasing interval: $\left(\frac{\pi }{2},\frac{3\pi }{2}\right)$
- $\cos x$ decreasing interval: $\left(0,\pi \right)$

The intersection of these two intervals is the region where both conditions are met. The common interval is:

$$\left(\frac{\pi }{2},\frac{3\pi }{2}\right)\cap \left(0,\pi \right)=\left(\frac{\pi }{2},\pi \right)$$

Conclusion

Both $\sin x$ and $\cos x$ are decreasing functions in the interval $\left(\frac{\pi }{2},\pi \right)$. This corresponds to option 2.