Solve the equation for exact solutions over the linterval .
7 tan
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is . Type an exact answer, using as needed. Type your answer in radians Use integers or fractio
for any numbers in the expression. Use a comma to separate answers as needed.)
B. The solution is the empty set.

Ask by Reyes Macdonald. in the United States

Nov 07,2024

Question

Solve the equation for exact solutions over the linterval .
7 tan
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is . Type an exact answer, using as needed. Type your answer in radians Use integers or fractio
for any numbers in the expression. Use a comma to separate answers as needed.)
B. The solution is the empty set.

Ask by Reyes Macdonald. in the United States

Nov 07,2024

UpStudy AI · Accepted Answer

To solve the equation $$ 7 \tan(3x) = 7 $$, we can first simplify it by dividing both sides by 7 (assuming $$ 7 \neq 0 $$):

$$
\tan(3x) = 1
$$

Next, we need to find the values of $$ 3x $$ for which $$ \tan(3x) = 1 $$. The tangent function equals 1 at the angles:

$$
3x = \frac{\pi}{4} + n\pi \quad \text{for } n \in \mathbb{Z}
$$

Now, we will solve for $$ x $$:

$$
x = \frac{\frac{\pi}{4} + n\pi}{3} = \frac{\pi}{12} + \frac{n\pi}{3}
$$

Next, we need to find the values of $$ x $$ in the interval $$ [0, 2\pi) $$. We will consider different integer values of $$ n $$:

1. For $$ n = 0 $$:
   $$
   x = \frac{\pi}{12}
   $$

2. For $$ n = 1 $$:
   $$
   x = \frac{\pi}{12} + \frac{\pi}{3} = \frac{\pi}{12} + \frac{4\pi}{12} = \frac{5\pi}{12}
   $$

3. For $$ n = 2 $$:
   $$
   x = \frac{\pi}{12} + \frac{2\pi}{3} = \frac{\pi}{12} + \frac{8\pi}{12} = \frac{9\pi}{12} = \frac{3\pi}{4}
   $$

4. For $$ n = 3 $$:
   $$
   x = \frac{\pi}{12} + \pi = \frac{\pi}{12} + \frac{12\pi}{12} = \frac{13\pi}{12}
   $$

5. For $$ n = 4 $$:
   $$
   x = \frac{\pi}{12} + \frac{5\pi}{3} = \frac{\pi}{12} + \frac{20\pi}{12} = \frac{21\pi}{12} = \frac{7\pi}{4}
   $$

6. For $$ n = 5 $$:
   $$
   x = \frac{\pi}{12} + 2\pi = \frac{\pi}{12} + \frac{24\pi}{12} = \frac{25\pi}{12} \quad \text{(not in } [0, 2\pi) \text{)}
   $$

Now we have the solutions in the interval $$ [0, 2\pi) $$:

$$
x = \frac{\pi}{12}, \frac{5\pi}{12}, \frac{3\pi}{4}, \frac{13\pi}{12}, \frac{7\pi}{4}
$$

Thus, the solution set is:

$$
\{ \frac{\pi}{12}, \frac{5\pi}{12}, \frac{3\pi}{4}, \frac{13\pi}{12}, \frac{7\pi}{4} \}
$$

So the correct choice is:

A. The solution set is $$ \{ \frac{\pi}{12}, \frac{5\pi}{12}, \frac{3\pi}{4}, \frac{13\pi}{12}, \frac{7\pi}{4} \} $$
The solution set is $$ \{ \frac{\pi}{12}, \frac{5\pi}{12}, \frac{3\pi}{4}, \frac{13\pi}{12}, \frac{7\pi}{4} \} $$.

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