the following equation involves multiple angles. solve the equation on the interval 0, 2π).\n\n\\( \\tan…

the following equation involves multiple angles. solve the equation on the interval 0, 2π).\n\n\\( \\tan \\frac { x } { 2 } = 1 \\)\n\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. x=\n(type an exact answer using π as needed. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)\n\nb. there is no solution.

the following equation involves multiple angles. solve the equation on the interval 0, 2π).\n\n\\( \\tan \\frac { x } { 2 } = 1 \\)\n\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. x=\n(type an exact answer using π as needed. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)\n\nb. there is no solution.

Answer

Explanation:

Step1: Solve for (\frac{x}{2})

We know that (\tan\theta = 1) when (\theta=\frac{\pi}{4}+k\pi), (k\in\mathbb{Z}). So for (\tan\frac{x}{2}=1), we have (\frac{x}{2}=\frac{\pi}{4}+k\pi).

Step2: Solve for (x)

Multiply both sides of (\frac{x}{2}=\frac{\pi}{4}+k\pi) by (2) to get (x = \frac{\pi}{2}+2k\pi).

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

When (k = 0), (x=\frac{\pi}{2}). When (k = 1), (x=\frac{\pi}{2}+2\pi=\frac{5\pi}{2}) (but (\frac{5\pi}{2}>2\pi)).

Answer:

A. (x=\frac{\pi}{2})