Animated Solution for Mathematics - Trigonometry: The number of solutions of $|\cos x|=\sin x$, such that $-4\pi \le x\le 4\pi$ is :

We are tasked with solving the equation $|\cos x|=\sin x$ within the domain $x\in [-4\pi ,4\pi ]$. This problem requires a careful balance of algebraic manipulation and geometric intuition.

The left-hand side involves an absolute value, which is inherently non-negative, i.e., $|\cos x|\ge 0$. Consequently, for the equation to hold, the right-hand side must satisfy the constraint:

This constraint acts as our "Gatekeeper." We are restricted to the intervals where the sine function is non-negative, specifically within the quadrants where the sine wave resides above the $x$-axis.

The sine function completes four full cycles within the interval $[-4\pi ,4\pi ]$. We must identify the specific sub-intervals where $\sin x\ge 0$:

1. In $[-4\pi ,-3\pi ]$, $\sin x\ge 0$. 2. In $[-2\pi ,-\pi ]$, $\sin x\ge 0$. 3. In $[0,\pi ]$, $\sin x\ge 0$. 4. In $[2\pi ,3\pi ]$, $\sin x\ge 0$.

These four intervals represent the regions where potential solutions exist. The periodic nature of these functions ensures that the behavior within each interval is identical.

Let us examine the interval $[0,\pi ]$ to determine the specific values of $x$. We split the equation $|\cos x|=\sin x$ into two cases based on the sign of $\cos x$:

Case 1: $\cos x\ge 0$. The equation simplifies to $\cos x=\sin x$, which implies $\tan x=1$. Within $[0,\pi ]$, this yields:

Case 2: $\cos x<0$. The equation simplifies to $-\cos x=\sin x$, which implies $\tan x=-1$. Within $[0,\pi ]$, this yields:

Thus, we have found exactly two solutions within the first valid interval. Because the functions $\sin x$ and $|\cos x|$ exhibit a periodic pattern, each of the four identified intervals will contribute exactly two solutions.

We have identified $4$ valid intervals, and each interval provides $2$ distinct solutions. The total number of solutions is calculated as:

By respecting the constraints of the absolute value and leveraging the periodicity of the trigonometric functions, we have successfully navigated the domain. The total number of solutions for the given equation is 8.