To solve the equation 4cos^2x + sinxcosx + 3sin^2x - 3 = 0, we can first rewrite it using trigonometric identities.
Since cos^2x + sin^2x = 1, we can rewrite the equation as:
4(1 - sin^2x) + sinx*cosx + 3sin^2x - 3 = 0.
Now, we can simplify it further:
4 - 4sin^2x + sinxcosx + 3sin^2x - 3 = 4 - sin^2x + sinxcosx = 0.
Next, we can use the identity sin 2x = 2sinxcosx to rewrite the equation:
4 - sin^2x + sinx*cosx = 4 - sin^2x + sin 2x / 2 = 4 - sin^2x + sin 2x / 2 = 4 - (2sinx)^2 + sin 2x / 2 = 4 - 2sin^2x + 2sinxcosx = 2(2 - sin^2x + sinxcosx) = 0.
Now, we can see that the equation is simplified to:
2(2 - sinx)(1 + sinx) = 0.
This equation can be solved by setting each factor equal to zero:
2 - sinx = 0 or 1 + sinx = sinx = 2 or sinx = -1.
However, the value of sin x cannot be 2, so we discard that and keep the value sinx = -1.
Therefore, the solution to the equation 4cos^2x + sinxcosx + 3sin^2x - 3 = 0 is sin x = -1.
To solve the equation 4cos^2x + sinxcosx + 3sin^2x - 3 = 0, we can first rewrite it using trigonometric identities.
Since cos^2x + sin^2x = 1, we can rewrite the equation as:
4(1 - sin^2x) + sinx*cosx + 3sin^2x - 3 = 0.
Now, we can simplify it further:
4 - 4sin^2x + sinxcosx + 3sin^2x - 3 =
4 - sin^2x + sinxcosx = 0.
Next, we can use the identity sin 2x = 2sinxcosx to rewrite the equation:
4 - sin^2x + sinx*cosx =
4 - sin^2x + sin 2x / 2 =
4 - sin^2x + sin 2x / 2 =
4 - (2sinx)^2 + sin 2x / 2 =
4 - 2sin^2x + 2sinxcosx =
2(2 - sin^2x + sinxcosx) = 0.
Now, we can see that the equation is simplified to:
2(2 - sinx)(1 + sinx) = 0.
This equation can be solved by setting each factor equal to zero:
2 - sinx = 0 or 1 + sinx =
sinx = 2 or sinx = -1.
However, the value of sin x cannot be 2, so we discard that and keep the value sinx = -1.
Therefore, the solution to the equation 4cos^2x + sinxcosx + 3sin^2x - 3 = 0 is sin x = -1.