Questions: Use a right triangle to write the following expression as an algebraic expression. Assume that x is positive and that the given inverse trigonometric function is defined for the expression in x. sec(cos^(-1) (2/x))

Solution Steps

To solve the given expression \(\sec \left(\cos ^{-1} \frac{2}{x}\right)\) using a right triangle, we need to follow these steps:

1. Recognize that \(\cos^{-1} \frac{2}{x}\) represents an angle \(\theta\) such that \(\cos \theta = \frac{2}{x}\).
2. Use the definition of cosine in a right triangle, where \(\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\).
3. Construct a right triangle where the adjacent side is 2 and the hypotenuse is \(x\).
4. Use the Pythagorean theorem to find the length of the opposite side.
5. Use the definition of secant, \(\sec \theta = \frac{1}{\cos \theta}\), to find the value of \(\sec \theta\).

We start by defining a right triangle where the angle \(\theta\) is such that \(\cos \theta = \frac{2}{x}\). In this triangle, we set the adjacent side to be 2 and the hypotenuse to be \(x\).

Using the Pythagorean theorem, we find the length of the opposite side: \[ \text{opposite} = \sqrt{x^2 - 2^2} = \sqrt{x^2 - 4} \]

The secant of the angle \(\theta\) is defined as: \[ \sec \theta = \frac{1}{\cos \theta} = \frac{x}{2} \] For \(x = 5\), we calculate: \[ \sec \theta = \frac{5}{2} = 2.5 \]

Final Answer