I can't figure out the answer to this problem
Use a right triangle to write the expression as an algebraic expression.
(Assume x is positive)
Tan(sec¯¹ [sq rt.x² + 4/x])
To approach this problem, let's break it down step by step.
1. Start by understanding the given expression: Tan(sec¯¹ [sq rt.x² + 4/x])
2. The expression inside the square brackets is √(x² + 4/x).
3. To simplify further, let's focus on the term x² + 4/x.
4. We observe that x² and 4/x are two different terms.
5. Let's consider a right triangle to help visualize this. In a right triangle, we have a hypotenuse, an opposite side, and an adjacent side.
6. Let's assign the following values to the sides of the right triangle:
- Hypotenuse: x
- Opposite side: √(x² + 4/x) (or let's say y)
- Adjacent side: 2/x (since 4/x can be simplified to 2/x)
7. Now, we can use trigonometric ratios to express the expression in the given problem. Since we have opposite and adjacent sides, the tangent (tan) ratio would be appropriate.
8. Recall that tan is defined as the ratio of the opposite side to the adjacent side: tan = opposite/adjacent.
9. In our triangle, tan = y / (2/x).
10. Simplifying this expression, we get tan = (x * y) / 2.
11. Therefore, the algebraic expression for the given problem is x * y / 2.
In summary, the expression Tan(sec¯¹ [sq rt.x² + 4/x]) can be written as the algebraic expression x * y / 2, where y = √(x² + 4/x).