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Given that [tex]\tan^2 \theta = \frac{3}{8}[/tex], what is the value of [tex]\sec \theta[/tex]?

A. [tex]\pm \sqrt{\frac{8}{3}}[/tex]

B. [tex]\pm \sqrt{\frac{11}{8}}[/tex]

C. [tex]\frac{11}{8}[/tex]

D. [tex]\frac{8}{3}[/tex]

Sagot :

GinnyAnswer
To find the value of [tex]\(\sec \theta\)[/tex] given that [tex]\(\tan^2 \theta = \frac{3}{8}\)[/tex], let's follow these steps:

1. Recall the trigonometric identity:
[tex]\[ \sec^2 \theta = 1 + \tan^2 \theta \][/tex]

2. Substitute the given value into the identity:
[tex]\[ \sec^2 \theta = 1 + \frac{3}{8} \][/tex]

3. Simplify the expression:
[tex]\[ \sec^2 \theta = 1 + \frac{3}{8} = \frac{8}{8} + \frac{3}{8} = \frac{11}{8} \][/tex]

4. To find [tex]\(\sec \theta\)[/tex], take the square root of both sides:
[tex]\[ \sec \theta = \pm \sqrt{\frac{11}{8}} \][/tex]

So, the value of [tex]\(\sec \theta\)[/tex] is [tex]\(\pm \sqrt{\frac{11}{8}}\)[/tex]. Among the given options, the correct answer is:
[tex]\[ \pm \sqrt{\frac{11}{8}} \][/tex]
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