Question

Writing to Learn Explain why there is a zero of \(y=\cos x\) between every two zeros of \(y=\sin x .\)

Short answer

Between any two successive zeros of the sine function, there is always a zero of the cosine function. This is due to the fact that zeros of the cosine function occur at odd multiples of \(\pi/2\), while zeros of the sine function occur at multiples of \(\pi\), hence there is always a zero of the cosine function between every two zeros of the sine function.

Step-by-step solution

Understanding sine and cosine functions

Both sine and cosine functions have the same range, from -1 to 1. The sine function starts from 0 at \(x = 0\) and goes to 1 at \(x = \frac{\pi}{2}\), then comes back to 0 at \(x = \pi\), then goes to -1 at \(x = \frac{3\pi}{2}\), and finally comes back to 0 at \(x = 2\pi\). The cosine function, on the other hand, starts from 1 at \(x = 0\), goes to 0 at \(x = \frac{\pi}{2}\), -1 at \(x = \pi\) and back to 0 at \( x = \frac{3\pi}{2}\), and finally it reaches 1 at \(x = 2\pi\)

Relating zeros of sine and cosine

From the previous observation, it can be seen that the zeros of the sine function occur at \(x = 0\), \(x = \pi\), \(x = 2\pi\), etc., i.e., at multiples of \(\pi\). The zeros of the cosine function occur at \(x = \frac{\pi}{2}\), \(x = \frac{3\pi}{2}\), etc., i.e., at odd multiples of \(\pi/2\).

Explaining the observation

Based on these observations, we see that between any two successive zeros of the sine function, there is always a zero of the cosine function. For example, between \(x = 0\) and \(x = \pi\) (where sine function has zeros), cosine function has zero at \(x = \frac{\pi}{2}\). Thus, it can be concluded that there is always a zero of \(y = \cos x\) between every two zeros of \(y = \sin x\).