Solve, finding all solutions in the interval [0, 2π). tan(3x) = -1

tan (3x) = -1

I know that 3x must be in quadrants II or IV
and I know that tan 45° = +1 or tan π/4 = 1
so 3x = π - π/4 OR 3x = 2π - π/4
3x = 3π/4 or 3x = 7π/4
x = π/4 or x = 7π/12

now the period of tan 3x is π/3
so adding multiples of π/3 to any answer will yield a new answer.

x = π/4
x = π/4 + π/3 = 7π/12 , we already have that one
x = 7π/12 + π/3 = 11π/12
x = 11π/12 + π/3 = 5π/4
x = 5π/4 + π/3 = 19π/12
x = 19π/12 + π/3 = 23π/12
The next one will be > 2π

So we have the above 6 solutions in the required domain.

btw, in degrees,
x = 45°, 105°, 165°, 225°, 285, and 345°

check two of them:
tan(3(105)) = -1
tan(3(345) = -1

To solve the equation tan(3x) = -1 in the interval [0, 2π), we can follow these steps:

Step 1: Find the principal solution
Start by finding the principal solution, which is the value of x that satisfies the equation without considering the interval. In this case, we have tan(3x) = -1. To find the principal solution, we need to take the inverse tangent of -1: arctan(-1).

Step 2: Find the general solution
The general solution is obtained by adding integer multiples of the period to the principal solution. Since the period of the tangent function is π (or 180 degrees), we can add π to the principal solution to find more solutions. In this case, the general solution is x = arctan(-1) + nπ, where n is an integer.

Step 3: Restrict the solutions within the given interval
Finally, we need to restrict the solutions to the interval [0, 2π). Check each value obtained from the general solution and keep only the solutions that fall within this interval.

Now let's find the solutions of the equation:

Step 1: Principal Solution
By taking the inverse tangent of -1: arctan(-1) ≈ -0.7854 radians or -45 degrees.

Step 2: General Solution
The general solution is x = arctan(-1) + nπ, where n is an integer. Plugging in the principal solution, we have x = -0.7854 + nπ.

Step 3: Restricting the solutions within [0, 2π)
We need to find n values that satisfy 0 ≤ -0.7854 + nπ < 2π. Let's solve the inequalities:
For the left side: 0 ≤ -0.7854 + nπ
Add 0.7854 to both sides and divide by π: 0.7854/π ≤ n
For the right side: -0.7854 + nπ < 2π
Add 0.7854 to both sides and divide by π: (2π + 0.7854)/π > n

We get the final solution for n as: 0.8273 ≤ n < 1.7854

Now we plug in all integer values of n in the range [0.8273, 1.7854) to find the solutions within the given interval.

Using n = 0: x = -0.7854 + 0π = -0.7854 radians
Using n = 1: x = -0.7854 + 1π ≈ 2.3562 radians

Therefore, the solutions in the interval [0, 2π) for the equation tan(3x) = -1 are approximately x = -0.7854 radians and x ≈ 2.3562 radians.

To solve the equation tan(3x) = -1 in the interval [0, 2π), we need to find all values of x that satisfy this condition.

Step 1: Find the principal solution

To find the principal solution, we can use the inverse tangent function (arctan). Taking the arctan of both sides of the equation, we get:

3x = arctan(-1)

Using a calculator, we find that arctan(-1) is equal to -π/4.

Therefore, 3x = -π/4.

Step 2: Find additional solutions

To find the additional solutions, we add integer multiples of the period of the tangent function, which is π.

To do this, we multiply both sides of the equation by 1/3:

x = (-π/4) * (1/3) = -π/12

Now, to find additional solutions, we add integer multiples of π to -π/12:

x = -π/12 + π = 11π/12

x = -π/12 + 2π = 23π/12

Since the interval given is [0, 2π), we only consider the solutions that fall within this interval:

x = -π/12 and x = 23π/12

Therefore, the solutions to the equation tan(3x) = -1 in the interval [0, 2π) are x = -π/12 and x = 23π/12.