Question

Решить: 7 tg x – 10ctg x + 9 = 0

Answer 1

Ответ:

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Answer 2

To solve the trigonometric equation 7tg(x) - 10ctg(x) + 9 = 0, we first need to simplify it.

Let's rewrite ctg(x) in terms of tg(x) using the identity: ctg(x) = 1/tg(x)

Now the equation becomes: 7tg(x) - 10/tg(x) + 9 = 0

To further simplify, let's find a common denominator for the terms involving tg(x). The common denominator is tg(x), so we can rewrite the equation as:

(7tg^2(x) - 10 + 9tg(x))/tg(x) = 0

Now, let's combine like terms:

(7tg^2(x) + 9tg(x) - 10)/tg(x) = 0

The equation is now in a form that we can solve more easily. To find the solutions, we set the numerator equal to zero:

7tg^2(x) + 9tg(x) - 10 = 0

Now, this is a quadratic equation in terms of tg(x). To solve for tg(x), we can use the quadratic formula:

tg(x) = [-b ± √(b^2 - 4ac)] / 2a

In our equation, a = 7, b = 9, and c = -10.

tg(x) = [-9 ± √(9^2 - 4 * 7 * -10)] / 2 * 7 tg(x) = [-9 ± √(81 + 280)] / 14 tg(x) = [-9 ± √(361)] / 14 tg(x) = [-9 ± 19] / 14

Now we have two possible values for tg(x):

1. tg(x) = (19 - 9) / 14 = 10/14 = 5/7
2. tg(x) = (-19 - 9) / 14 = -28/14 = -2

Next, we can find the values of x using the inverse tangent function (arctan or tan^(-1)):

1. x = arctan(5/7) ≈ 37.5 degrees (rounded to one decimal place).
2. x = arctan(-2) ≈ -63.4 degrees (rounded to one decimal place).

Please note that trigonometric functions have periodicity, and there are infinite solutions to the given equation. The above solutions represent one period of the function. To find all solutions, you can add multiples of the period (180 degrees or π radians) to each of the above solutions.