To simplify this expression, we can recognize that it can be rewritten as:
3sin^2x + 14sinxcosx + 8cos^2x = 0
Using the trigonometric identity sin^2x + cos^2x = 1, we can substitute cos^2x = 1 - sin^2x:
3sin^2x + 14sinx(1 - sin^2x) + 8(1 - sin^2x) = 0
Expand the expression:
3sin^2x + 14sinx - 14sin^3x + 8 - 8sin^2x = 0
Rearrange and combine like terms:
-14sin^3x + 3sin^2x - 8sinx + 8 = 0
This is a cubic equation in sin(x), and it can be solved using various methods such as factoring, graphing, or numerical methods.
To simplify this expression, we can recognize that it can be rewritten as:
3sin^2x + 14sinxcosx + 8cos^2x = 0
Using the trigonometric identity sin^2x + cos^2x = 1, we can substitute cos^2x = 1 - sin^2x:
3sin^2x + 14sinx(1 - sin^2x) + 8(1 - sin^2x) = 0
Expand the expression:
3sin^2x + 14sinx - 14sin^3x + 8 - 8sin^2x = 0
Rearrange and combine like terms:
-14sin^3x + 3sin^2x - 8sinx + 8 = 0
This is a cubic equation in sin(x), and it can be solved using various methods such as factoring, graphing, or numerical methods.