What are the [tex]\( x \)[/tex]-intercepts of the graph of the function [tex]\( y = 6 \tan \left( \frac{x}{2} \right) - 3 \)[/tex]?

A. [tex]\( (0.9273 + n\pi, 0) \)[/tex], where [tex]\( n \)[/tex] is any integer
B. [tex]\( (0.9653 + n\pi, 0) \)[/tex], where [tex]\( n \)[/tex] is any integer
C. [tex]\( (0.9273 + 2n\pi, 0) \)[/tex], where [tex]\( n \)[/tex] is any integer
D. [tex]\( (0.9653 + 2n\pi, 0) \)[/tex], where [tex]\( n \)[/tex] is any integer



Answer :

To find the [tex]$x$[/tex]-intercepts of the graph of the function [tex]\( y = 6 \tan \left( \frac{x}{2} \right) - 3 \)[/tex], we need to determine where the function crosses the [tex]\( x \)[/tex]-axis. In other words, we need to solve for [tex]\( x \)[/tex] when [tex]\( y = 0 \)[/tex].

1. Start with the equation:
[tex]\[ 6 \tan \left( \frac{x}{2} \right) - 3 = 0 \][/tex]

2. Isolate the tangent function by adding 3 to both sides:
[tex]\[ 6 \tan \left( \frac{x}{2} \right) = 3 \][/tex]

3. Divide both sides by 6:
[tex]\[ \tan \left( \frac{x}{2} \right) = \frac{1}{2} \][/tex]

4. To solve for [tex]\( x \)[/tex], consider the equation [tex]\( \tan \left( \frac{x}{2} \right) = \frac{1}{2} \)[/tex]. The tangent function has a period of [tex]\( \pi \)[/tex], so the general solution is:
[tex]\[ \frac{x}{2} = \arctan \left( \frac{1}{2} \right) + n\pi \quad \text{for any integer } n \][/tex]

5. Isolate [tex]\( x \)[/tex] by multiplying both sides by 2:
[tex]\[ x = 2 \arctan \left( \frac{1}{2} \right) + 2n\pi \][/tex]

6. The principal value of [tex]\( \arctan \left( \frac{1}{2} \right) \)[/tex] is approximately [tex]\( 0.4636 \)[/tex]. Therefore:
[tex]\[ 2 \arctan \left( \frac{1}{2} \right) \approx 2 \times 0.4636 \approx 0.9273 \][/tex]

7. Hence, the [tex]\( x \)[/tex]-intercepts of the function are:
[tex]\[ x = 0.9273 + 2n\pi \quad \text{for any integer } n \][/tex]

So, the correct form of the [tex]\( x \)[/tex]-intercepts is:

[tex]\[ (0.9273 + 2n \pi, 0) \quad \text{where } n \text{ is any integer} \][/tex]

Thus, the correct answer is:

[tex]\[ (0.9273 + 2n \pi, 0) \quad \text{where } n \text{ is any integer} \][/tex]