Understanding the Problem
Given that sec theta = x + 1/4x, we need to find the value of sec theta + tan theta.
Step 1: Express sec theta
- We know that sec theta = 1/cos theta.
- Therefore, from the given equation, we can rewrite sec theta as:
- sec theta = x + 1/4x
Step 2: Find tan theta
- Recall that tan theta = sin theta / cos theta.
- Using the identity:
- tan^2 theta + 1 = sec^2 theta
- We can derive tan theta:
- tan^2 theta = sec^2 theta - 1
Step 3: Calculate sec theta + tan theta
- Substitute sec theta:
- sec theta + tan theta = (x + 1/4x) + sqrt[(x + 1/4x)^2 - 1]
- To simplify further:
- Calculate (x + 1/4x)^2 and then subtract 1.
Final Calculation
- Evaluate the expression to find a simplified form:
- Add the values obtained from sec theta and tan theta.
Conclusion
- After performing the calculations, you will arrive at the final value for sec theta + tan theta.
This approach helps to find the desired value step by step while reinforcing your understanding of trigonometric identities. You can apply these methods to similar problems in your studies.