Animated Solution for Mathematics - Trigonometry: If the sum of solutions of the system of equations $2{\sin }^2\theta -\cos 2\theta =0$ and $2{\cos }^2\theta +3\sin \theta =0$ in the interval $[0,2\pi ]$ is $k\pi$, then $k$ is equal to ______.

The objective is to find the values of $\theta$ in the interval $[0,2\pi ]$ that satisfy a system of two trigonometric equations. We seek the harmony between these two constraints to identify the intersection of their solution sets.

Unifying the First Equation

We begin with the first equation:

To solve this, we utilize the double-angle identity $\cos 2\theta =1-2{\sin }^2\theta$. Substituting this into the equation, we obtain:

Simplifying the expression leads to:

Taking the square root, we find $\sin \theta =\pm \frac{1}{2}$. This yields four potential angles in the interval $[0,2\pi ]$:

The Second Constraint

Next, we address the second equation:

Using the Pythagorean identity ${\cos }^2\theta =1-{\sin }^2\theta$, we substitute to unify the variables:

Expanding and rearranging into standard quadratic form, we get:

Factoring the quadratic expression, we identify the roots:

This results in two cases: $\sin \theta =-\frac{1}{2}$ or $\sin \theta =2$. Since $\sin \theta =2$ is impossible, we focus on $\sin \theta =-\frac{1}{2}$, which occurs at:

The Intersection

The system requires that both equations be satisfied simultaneously. We compare the solution sets:

The intersection $S_1\cap S_2$ is clearly:

The problem asks for the sum of these solutions. Adding them together:

Since the sum is represented as $k\pi$, we conclude that $k=3$.