Question

3sin^2 (3x)-sin (3x)-4=0

Answer 1

Сделаем замену: t = sin(3x)
3t^2 - t - 4 = 0
D = 1 + 48 = 49 корень(D) = 7
t1 = (1+7)/6 = 4/3
t2 = (1-7)/6 = -1
sin(3x) = 4/3 - нет решения, т.к. синус не превосходит единицы.
sin(3x) = -1
3x = -p/2 + n*p
x = -p/6 + n*p/3

Answer 2

To solve the equation 3sin^2(3x) - sin(3x) - 4 = 0, we can use a substitution to simplify the equation. Let's denote y = sin(3x).

Substituting y into the equation, we have: 3y^2 - y - 4 = 0

Now, we can solve this quadratic equation for y. We can factorize or use the quadratic formula to find the solutions.

Factoring: (3y + 4)(y - 1) = 0

Setting each factor equal to zero: 3y + 4 = 0 or y - 1 = 0

Solving for y: 3y = -4 or y = 1

Dividing both sides by 3 in the first equation: y = -4/3

Now, we substitute y back into the equation y = sin(3x):

For y = -4/3: sin(3x) = -4/3

For y = 1: sin(3x) = 1

To find the values of x, we need to take the inverse sine (arcsin) of both sides. Keep in mind that the inverse sine function has multiple solutions, so we'll need to find all possible values within a given range.

For sin(3x) = -4/3: No solution exists because the range of the sine function is -1 to 1, so -4/3 is outside this range.

For sin(3x) = 1: 3x = π/2 + 2πn (n is an integer)

Solving for x: x = (π/6 + 2πn)/3 (n is an integer)

Therefore, the solutions to the equation 3sin^2(3x) - sin(3x) - 4 = 0 are given by: x = (π/6 + 2πn)/3 (n is an integer)