In Exercises 83–94, use a right triangle to write each 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 (sin⁻¹ x/√x²+4)

Verified step by step guidance

1. Step 1: Recognize that \( \sin^{-1} \left( \frac{x}{\sqrt{x^2 + 4}} \right) \) represents an angle \( \theta \) such that \( \sin \theta = \frac{x}{\sqrt{x^2 + 4}} \).
2. Step 2: Construct a right triangle where the opposite side to angle \( \theta \) is \( x \) and the hypotenuse is \( \sqrt{x^2 + 4} \).
3. Step 3: Use the Pythagorean theorem to find the adjacent side: \( \text{adjacent} = \sqrt{(\sqrt{x^2 + 4})^2 - x^2} = \sqrt{x^2 + 4 - x^2} = \sqrt{4} = 2 \).
4. Step 4: Calculate \( \sec \theta \), which is the reciprocal of \( \cos \theta \). Since \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{2}{\sqrt{x^2 + 4}} \), then \( \sec \theta = \frac{\sqrt{x^2 + 4}}{2} \).
5. Step 5: Express \( \sec(\sin^{-1}(\frac{x}{\sqrt{x^2 + 4}})) \) as the algebraic expression \( \frac{\sqrt{x^2 + 4}}{2} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as sin⁻¹ (arcsine), are used to find angles when the value of a trigonometric function is known. For example, sin⁻¹(x) gives the angle whose sine is x. Understanding how to interpret these functions is crucial for solving problems involving angles and their relationships in right triangles.

Right Triangle Relationships

In a right triangle, the relationships between the angles and sides are defined by trigonometric ratios: sine, cosine, and tangent. For instance, if θ is an angle, then sin(θ) = opposite/hypotenuse and cos(θ) = adjacent/hypotenuse. These relationships allow us to express trigonometric functions in terms of the triangle's sides, which is essential for converting expressions involving inverse functions.

Secant Function

The secant function, denoted as sec(θ), is the reciprocal of the cosine function, defined as sec(θ) = 1/cos(θ). In the context of a right triangle, sec(θ) can be expressed in terms of the triangle's sides, specifically as sec(θ) = hypotenuse/adjacent. Understanding how to manipulate and express secant in terms of other trigonometric functions is key to solving the given expression.