Question

Find number of solutions of the equation $\left|\tan (|2x|+\frac{\pi }{3})\right|=3$ in $(-\pi ,\pi )$

• A. 6
• B. 8
• C. 10
• D. 4

Answer 1

The correct option is B 8

As we know, number of solutions of equation $\left|\tan (|2x|+\frac{\pi }{3})\right|=3$ in $(-\pi ,\pi )$ is same as the number of points of intersection of $y=\left|\tan (|2x|+\frac{\pi }{3})\right|$ and $y=3$ in $(-\pi ,\pi )$.
Let's draw the graph of $y=\left|\tan (|2x|+\frac{\pi }{3})\right|$.
We can see, fundamental function involved here is $\tan x$ whose graph is given by

Now apply the shrink along $x$-axis to $y=\tan x$ to get $y=\tan 2x$ as shown below

Further apply the modulus transformation to get $y=\tan |2x|$ as shown below,


Now apply the horizontal shift to get $y=\tan (|2x|+\frac{\pi }{3})$ as shown below,
Here we need to apply modulus flip along x-axis and we get the graph of $y=\left|\tan (|2x|+\frac{\pi }{3})\right|$ as shown below



Finally we need to draw $y=3$ and check the number of points of intersection in the graph

We can see that there are 8 points of intersection in $(-\pi ,\pi )$.
Hence the given equation have 8 solutions in $(-\pi ,\pi )$.