Question

4sin3x+cos3x=4

Answer 1

2(cos4x - sinx*cos3x) = sin4x + sin2x 
2*cos4x - sin4x + sin2x = sin4x + sin2x (*) 
cos4x - sin4x = 0 
cos4x = sin4x 
4x = Pi/4 + Pi*k 
x = Pi/8 + Pi/4 * k

Answer 2

To solve the equation 4sin(3x) + cos(3x) = 4, we can use trigonometric identities and algebraic techniques. Let's break it down step by step:

1. Start by expressing sin(3x) and cos(3x) in terms of sin(x) and cos(x) using the triple-angle identities: sin(3x) = 3sin(x) - 4sin^3(x) cos(3x) = 4cos^3(x) - 3cos(x)

2. Substitute these expressions into the equation: 4(3sin(x) - 4sin^3(x)) + 4cos^3(x) - 3cos(x) = 4

3. Simplify the equation: 12sin(x) - 16sin^3(x) + 4cos^3(x) - 3cos(x) = 4

4. Rearrange the terms: -16sin^3(x) + 4cos^3(x) + 12sin(x) - 3cos(x) = 4

5. Combine like terms: -16sin^3(x) + 4cos^3(x) + 12sin(x) - 3cos(x) - 4 = 0

6. Now, we can attempt to factor the equation. However, it's important to note that this equation might not have a simple factorization or an exact algebraic solution. In such cases, numerical or iterative methods may be required to find an approximate solution.

If you're looking for an exact solution, it might be necessary to use numerical methods or a graphing calculator to approximate the values of x that satisfy the equation.