Question

Tgx x tg3x = -1 please

Answer 1

Очевидно что
$x = \frac{\pi}{4} + \pi \: k$
где k - любое целое число.
Решено методом подбора. Догадаться не составит большого труда и вам.

Answer 2

To solve the equation tg(x) * tg(3x) = -1, we can use the trigonometric identity:

tan(A) * tan(B) = (sin(A) / cos(A)) * (sin(B) / cos(B)) = (sin(A) * sin(B)) / (cos(A) * cos(B))

In this case, A = x and B = 3x. Therefore, we have:

tan(x) * tan(3x) = (sin(x) * sin(3x)) / (cos(x) * cos(3x))

Now, we need to simplify this expression. We can use the identities:

sin(3x) = 3sin(x) - 4sin^3(x) cos(3x) = 4cos^3(x) - 3cos(x)

Substituting these values into the equation, we have:

tan(x) * tan(3x) = (sin(x) * (3sin(x) - 4sin^3(x))) / (cos(x) * (4cos^3(x) - 3cos(x)))

Expanding further, we get:

tan(x) * tan(3x) = (3sin^2(x) - 4sin^4(x)) / (4cos^2(x) - 3cos^4(x))

Now, we need to solve for x when this expression equals -1. However, this equation involves fourth powers of trigonometric functions, which makes it difficult to solve analytically. It may require numerical methods or approximations to find the solutions.