2sin^2x-4cos^2x=7sinx×cosx

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вопрос

To solve the equation 2sin^2x - 4cos^2x = 7sinx * cosx, we can use the trigonometric identity sin^2(x) + cos^2(x) = 1. This allows us to express sin^2(x) in terms of cos^2(x) and rewrite the equation as:

2(1 - cos^2(x)) - 4cos^2(x) = 7sinx cosx
2 - 2cos^2(x) - 4cos^2(x) = 7sinx cosx
2 - 6cos^2(x) = 7sinx * cosx

Now we can express sinx and cosx in terms of tan(x):

sinx = tanx / √(1 + tan^2(x))
cosx = 1 / √(1 + tan^2(x))

Substitute these into the equation:

2 - 6(1 / (1 + tan^2(x))) = 7(tanx / √(1 + tan^2(x)) * (1 / √(1 + tan^2(x)))
2 - 6 / (1 + tan^2(x)) = 7tan^2(x) / (1 + tan^2(x))

Now we have a quadratic equation in terms of tan(x). We can simplify this equation and solve for tan(x) using the quadratic formula. Once we have found tan(x), we can find sin(x) and cos(x) using the identities above.