Identify the general solution to the equation: 3(tan(2x) - 4) + 9 = 0.

The given equation is

3(tan(2x) - 4) + 9 = 0

First, distribute the 3 and combine the like terms:

3tan(2x) − 12 + 9 = 0

3tan(2x) − 3 = 0

Add 3 to both sides and divide both sides by 3

3tan(2x) = 3

tan(2x) = 1

To solve for 2x, take the arctangent (inverse tangent) of both sides:

2x = arctan(1)

Since \$ arctan(1) = pi/4 + npi \$ where n is any integer, we have:

\$ 2x = pi/4 + npi \$

Finally, solve for x by dividing both sides by 2

\$ x = pi/8 + (npi)/2 \$

So, the general solution for x is \$ x = pi/8 + (npi)/2 \$, where n is any integer.

It represents a solution where x can take on any value equal to π plus an integer multiple of π. This doesn’t match the general solution we derived, so option A is incorrect

\$ x = pi + npi\$

This option suggests that x equals \$pi/2\$ plus an integer multiple of \$(3pi)/2\$. it doesn’t match the general solution we derived, so option B is incorrect

\$ x = pi/2 + (3npi)/2\$

This option matches the general solution we derived: x = \$ pi/8 + (npi)/2\$, where n is an integer. It correctly represents the general solution to the equation, so option C is correct

\$ x = pi/8 + (npi)/2\$

This option suggests that x equals \$pi/3\$ plus an integer multiple of \$pi/3\$. It doesn’t match the general solution we derived, so option D is incorrect

\$ x = pi/3 + (npi)/3\$

Option

C

Can you summarize what you’ve understood in the above steps?