To solve this equation, we first need to expand the left side of the equation:
Sinx(4 sinx - 1) = 4sin^2x - sinx
Next, we substitute this expansion back into the original equation:
4sin^2x - sinx = 2 + 3√cosx
Now we need to convert sinx and cosx into the same trigonometric function so that we can solve the equation. We know that sin^2x + cos^2x = 1, so cosx = 1 - sin^2x.
Substitute cosx = 1 - sin^2x into the equation:
4sin^2x - sinx = 2 + 3√(1 - sin^2x)
Now, isolate the square root term on one side of the equation:
4sin^2x - sinx - 2 = 3√(1 - sin^2x)
Square both sides of the equation to eliminate the square root:
This is a quartic equation in terms of sinx. Solving this equation for sinx involves finding the roots using methods such as factoring, the quadratic formula, or numerical methods.
To solve this equation, we first need to expand the left side of the equation:
Sinx(4 sinx - 1) = 4sin^2x - sinx
Next, we substitute this expansion back into the original equation:
4sin^2x - sinx = 2 + 3√cosx
Now we need to convert sinx and cosx into the same trigonometric function so that we can solve the equation. We know that sin^2x + cos^2x = 1, so cosx = 1 - sin^2x.
Substitute cosx = 1 - sin^2x into the equation:
4sin^2x - sinx = 2 + 3√(1 - sin^2x)
Now, isolate the square root term on one side of the equation:
4sin^2x - sinx - 2 = 3√(1 - sin^2x)
Square both sides of the equation to eliminate the square root:
(4sin^2x - sinx - 2)^2 = 9(1 - sin^2x)
Expand both sides:
16sin^4x - 8sin^3x - 16sin^2x + 8sinx + 4 - sin^2x = 9 - 9sin^2x
Combine like terms:
16sin^4x - 8sin^3x - 17sin^2x + 8sinx - 5 = 9 - 9sin^2x
Rearrange the terms:
16sin^4x - 8sin^3x - 8sin^2x - 8sinx - 14 = 0
This is a quartic equation in terms of sinx. Solving this equation for sinx involves finding the roots using methods such as factoring, the quadratic formula, or numerical methods.