At Westonci.ca, we provide clear, reliable answers to all your questions. Join our vibrant community and get the solutions you need. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
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]
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]
We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.