solve the following equation on the interval 0,2π). tan(3x)=1 select the correct choice below and, if…

solve the following equation on the interval 0,2π). tan(3x)=1 select the correct choice below and, if necessary, fill in the answer box to complete your choice. oa. x= (type an exact answer, using π as needed. use a comma to separate answers as needed. type your answer in the expression.) ob. there is no solution.

solve the following equation on the interval 0,2π). tan(3x)=1 select the correct choice below and, if necessary, fill in the answer box to complete your choice. oa. x= (type an exact answer, using π as needed. use a comma to separate answers as needed. type your answer in the expression.) ob. there is no solution.

Answer

Explanation:

Step1: Find the general solution for (3x)

We know that if (\tan\theta = 1), then (\theta=\frac{\pi}{4}+k\pi), (k\in\mathbb{Z}). For (\tan(3x) = 1), we set (3x=\frac{\pi}{4}+k\pi).

Step2: Solve for (x)

Divide both sides of the equation (3x=\frac{\pi}{4}+k\pi) by (3). We get (x = \frac{\pi}{12}+\frac{k\pi}{3}).

Step3: Find solutions in the interval ([0,2\pi))

When (k = 0): (x=\frac{\pi}{12}) When (k = 1): (x=\frac{\pi}{12}+\frac{\pi}{3}=\frac{\pi + 4\pi}{12}=\frac{5\pi}{12}) When (k = 2): (x=\frac{\pi}{12}+\frac{2\pi}{3}=\frac{\pi+8\pi}{12}=\frac{9\pi}{12}=\frac{3\pi}{4}) When (k = 3): (x=\frac{\pi}{12}+\pi=\frac{\pi + 12\pi}{12}=\frac{13\pi}{12}) When (k = 4): (x=\frac{\pi}{12}+\frac{4\pi}{3}=\frac{\pi + 16\pi}{12}=\frac{17\pi}{12}) When (k = 5): (x=\frac{\pi}{12}+\frac{5\pi}{3}=\frac{\pi+20\pi}{12}=\frac{21\pi}{12}=\frac{7\pi}{4}) When (k = 6): (x=\frac{\pi}{12}+2\pi=\frac{\pi + 24\pi}{12}=\frac{25\pi}{12}>2\pi) (rejected)

Answer:

A. (x = \frac{\pi}{12},\frac{5\pi}{12},\frac{3\pi}{4},\frac{13\pi}{12},\frac{17\pi}{12},\frac{7\pi}{4})