Question

4sin^2x+3sinx*cosx-7cos^2x=0

Answer 1

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Answer 2

The given equation is a trigonometric equation involving both sine and cosine functions. The goal is to solve for the values of x that satisfy the equation. Let's work through the steps to solve it:

The given equation is: $4\sin^2(x) + 3\sin(x)\cos(x) - 7\cos^2(x) = 0.$

To simplify the equation, we can use the trigonometric identity $\sin^2(x) + \cos^2(x) = 1$ to replace $\sin^2(x)$ and $\cos^2(x)$ in terms of each other: $4(1 - \cos^2(x)) + 3\sin(x)\cos(x) - 7\cos^2(x) = 0.$

Now distribute the constants: $4 - 4\cos^2(x) + 3\sin(x)\cos(x) - 7\cos^2(x) = 0.$

Combine the terms involving $\cos^2(x)$: $4 - 11\cos^2(x) + 3\sin(x)\cos(x) = 0.$

Now, we can use a trigonometric identity $\sin(2x) = 2\sin(x)\cos(x)$ to replace $3\sin(x)\cos(x)$ with $1.5\sin(2x)$: $4 - 11\cos^2(x) + 1.5\sin(2x) = 0.$

Rearrange the equation: $11\cos^2(x) - 1.5\sin(2x) - 4 = 0.$

At this point, finding a general solution for $x$ may not be straightforward due to the presence of both sine and cosine terms. Depending on the context of the problem, you might need to resort to numerical methods or specific techniques to approximate the solutions.

If you have specific numerical values for $x$ that you're trying to solve for, you can use numerical methods such as the bisection method, Newton's method, or a calculator with equation-solving capabilities to find approximate solutions.

If you're looking for a more symbolic solution or further analysis, please provide additional context or constraints, and I'll do my best to assist you.