Question

(Sin x)^2-2sinx cosx=3(cosx)^2

Answer 1

Данное уравнение является однородным. Идея решения заключается в следующем: разделим обе части уравнения на (cos(x))^2 (cos(x)=0 не является корнем этого уравнения, поэтому мы имеем право делить). Получаем квадратное уравнение относительно tg(x).

Полное решение во вложении.

Answer 2

To solve the equation (sin x)^2 - 2sin x cos x = 3(cos x)^2, we can use trigonometric identities to simplify it. Let's go step by step:

Step 1: Use the Pythagorean identity for sine and cosine: (sin x)^2 + (cos x)^2 = 1

Step 2: Divide the entire equation by (cos x)^2 to make the substitution for (sin x)^2:

(sin x)^2/cos^2 x - 2(sin x)/cos x = 3

Step 3: Now, we can substitute (sin x)^2 with (1 - (cos x)^2):

(1 - (cos x)^2)/cos^2 x - 2(sin x)/cos x = 3

Step 4: Simplify the equation:

(1/cos^2 x) - (cos x)^2/cos^2 x - 2(sin x)/cos x = 3

Step 5: Combine the terms:

1/cos^2 x - (cos x)^2/cos^2 x - 2(sin x)/cos x = 3

Step 6: Get a common denominator:

1/cos^2 x - (cos x)^2/cos^2 x - (2sin x * cos x)/cos^2 x = 3

Step 7: Combine the terms again:

(1 - (cos x)^2 - 2sin x * cos x)/cos^2 x = 3

Step 8: Use the Pythagorean identity again to simplify the numerator:

(1 - cos^2 x - 2sin x * cos x)/cos^2 x = 3

Now, the expression in the numerator can be simplified further using the identity 1 - cos^2 x = sin^2 x:

(sin^2 x - 2sin x * cos x)/cos^2 x = 3

Step 9: Factor out sin^2 x from the numerator:

(sin^2 x * (1 - 2cos x))/cos^2 x = 3

Step 10: Divide both sides by cos^2 x:

(sin^2 x * (1 - 2cos x)) = 3 * cos^2 x

Step 11: Now, use the identity sin^2 x = 1 - cos^2 x:

((1 - cos^2 x) * (1 - 2cos x)) = 3 * cos^2 x

Step 12: Expand the left side of the equation:

(1 - 3cos^2 x + 2cos^3 x) = 3 * cos^2 x

Step 13: Move all terms to one side to obtain a cubic equation:

2cos^3 x - 4cos^2 x + 1 = 0

This is a cubic equation in terms of cos x. Solving cubic equations analytically can be quite involved and may not have simple solutions. However, numerical methods or approximations can be used to find solutions for specific values of x.

If you were given specific constraints or a range for the variable x, then you could use numerical methods like the Newton-Raphson method or graphical methods to find approximate solutions. Otherwise, the equation in terms of cos x is as far as we can simplify it algebraically.