Question

4tg x-12ctg x + 13=0

Answer 1

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Answer 2

The equation you've provided is a trigonometric equation. To solve it, we'll first simplify the equation using trigonometric identities and then find the values of x that satisfy the equation.

Let's work through the steps:

1. Start with the equation: 4tg(x) - 12ctg(x) + 13 = 0.

2. To simplify the equation, we can use the identity: ctg(x) = 1/tg(x).

   Applying this identity, the equation becomes: 4tg(x) - 12/tg(x) + 13 = 0.

3. Next, let's find a common denominator by multiplying the equation by tg(x):

   4(tg(x))^2 - 12 + 13(tg(x)) = 0.

4. Now, let's simplify further:

   4(tg(x))^2 + 13(tg(x)) - 12 = 0.

5. We can treat this equation as a quadratic equation in terms of tg(x). Let's substitute a variable, let's say "t," for tg(x):

   4t^2 + 13t - 12 = 0.

6. We can now solve this quadratic equation using factoring, completing the square, or the quadratic formula. In this case, let's use the quadratic formula:

   t = (-b ± √(b^2 - 4ac)) / (2a),

   where a = 4, b = 13, and c = -12.

   Plugging in these values:

   t = (-13 ± √(13^2 - 4 * 4 * -12)) / (2 * 4).

7. Simplifying the expression inside the square root:

   t = (-13 ± √(169 + 192)) / 8.

   t = (-13 ± √(361)) / 8.

   t = (-13 ± 19) / 8.

8. Splitting it into two solutions:

   Solution 1: t = (-13 + 19) / 8 = 6 / 8 = 3 / 4.

   Solution 2: t = (-13 - 19) / 8 = -32 / 8 = -4.

9. Now, recall that t = tg(x). We can find x by taking the inverse tangent (arctan) of the solutions:

   Solution 1: x = arctan(3/4).

   Solution 2: x = arctan(-4).

These are the solutions for x in the given equation. Note that the trigonometric functions are periodic, so there are infinitely many solutions. The solutions we found represent the principal values within a specific range.