7 tg x – 12ctg x + 8 = 0

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To solve this trigonometric equation, we can first rewrite the cotangent function in terms of tangent:

ctg(x) = 1/tg(x)

So our equation becomes:

7tg(x) - 12(1/tg(x)) + 8 = 0

Multiplying through by tg(x) to clear the denominator:

7(tg(x))^2 - 12 + 8tg(x) = 0

Rearranging terms:

7(tg(x))^2 + 8tg(x) - 12 = 0

Now we have a quadratic equation in terms of tg(x). Let's substitute y = tg(x) to make it easier to solve:

7y^2 + 8y - 12 = 0

We can solve this quadratic equation using the quadratic formula:

y = [-b ± sqrt(b^2 - 4ac)] / 2a

In this case, a = 7, b = 8, and c = -12. Substituting into the formula:

y = [-8 ± sqrt(8^2 - 4(7)(-12))] / 2(7)
y = [-8 ± sqrt(64 + 336)] / 14
y = [-8 ± sqrt(400)] / 14
y = [-8 ± 20] / 14

Therefore, the solutions for y are:

y1 = (20 - 8) / 14 = 12 / 14 = 6 / 7

y2 = (-20 - 8) / 14 = -28 / 14 = -2

Now, we can find the solutions for x by setting y = tg(x):

y1 = tg(x) = 6/7
x = arctan(6/7)

y2 = tg(x) = -2
x = arctan(-2)

So the solutions to the equation 7tg(x) – 12ctg(x) + 8 = 0 are x = arctan(6/7) and x = arctan(-2).