To find the value of [tex]\(\sec \theta\)[/tex], given that [tex]\(\tan^2 \theta = \frac{3}{8}\)[/tex], let's follow a step-by-step solution:
1. Identify the given information:
[tex]\(\tan^2 \theta = \frac{3}{8}\)[/tex]
2. Calculate [tex]\(\tan \theta\)[/tex]:
[tex]\[\tan \theta = \sqrt{\tan^2 \theta} = \sqrt{\frac{3}{8}}\][/tex]
3. Use the trigonometric identity involving sec and tan:
Recall that the identity [tex]\(\sec^2 \theta = 1 + \tan^2 \theta\)[/tex] relates secant and tangent.
4. Substitute [tex]\(\tan^2 \theta\)[/tex] into the identity:
[tex]\[\sec^2 \theta = 1 + \frac{3}{8}\][/tex]
5. Simplify the expression:
[tex]\[1 + \frac{3}{8} = \frac{8}{8} + \frac{3}{8} = \frac{11}{8}\][/tex]
Thus, we find:
[tex]\[\sec^2 \theta = \frac{11}{8}\][/tex]
6. Solve for [tex]\(\sec \theta\)[/tex]:
[tex]\[\sec \theta = \pm \sqrt{\sec^2 \theta} = \pm \sqrt{\frac{11}{8}}\][/tex]
Therefore, the value of [tex]\(\sec \theta\)[/tex] is [tex]\(\pm \sqrt{\frac{11}{8}}\)[/tex].
So, the correct answer is:
[tex]\(\pm \sqrt{\frac{11}{8}}\)[/tex]
