In Exercises 52–53, 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 1/x)

Verified step by step guidance

1. Recognize that the expression is \( \sec(\sin^{-1}(1/x)) \). Let \( \theta = \sin^{-1}(1/x) \), which means \( \sin(\theta) = \frac{1}{x} \) and \( \theta \) is an angle in a right triangle.
2. Draw a right triangle where the angle \( \theta \) has an opposite side of length 1 and a hypotenuse of length \( x \), since \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{x} \).
3. Use the Pythagorean theorem to find the adjacent side of the triangle: \( \text{adjacent} = \sqrt{x^2 - 1^2} = \sqrt{x^2 - 1} \).
4. Recall that \( \sec(\theta) = \frac{\text{hypotenuse}}{\text{adjacent}} \). Substitute the known sides to write \( \sec(\sin^{-1}(1/x)) = \frac{x}{\sqrt{x^2 - 1}} \).
5. Express the final algebraic expression for \( \sec(\sin^{-1}(1/x)) \) as \( \frac{x}{\sqrt{x^2 - 1}} \), which is valid for \( x > 1 \) to keep the expression defined.

Key Concepts

Inverse Sine Function (sin⁻¹ or arcsin)

The inverse sine function, sin⁻¹(y), returns the angle whose sine is y. It is defined for inputs between -1 and 1, and its output angle lies between -π/2 and π/2. Understanding this helps to interpret sin⁻¹(1/x) as an angle in a right triangle where the opposite side over hypotenuse equals 1/x.

Right Triangle Trigonometry

Right triangle trigonometry relates the sides of a right triangle to its angles using sine, cosine, and secant functions. By representing sin⁻¹(1/x) as an angle θ, you can construct a triangle with opposite side 1 and hypotenuse x, then use the Pythagorean theorem to find the adjacent side and express sec(θ) algebraically.

Secant Function (sec)

The secant function is the reciprocal of cosine: sec(θ) = 1/cos(θ). After finding the angle θ = sin⁻¹(1/x), sec(θ) can be expressed in terms of the triangle's sides as hypotenuse over adjacent side. This allows rewriting sec(sin⁻¹(1/x)) as an algebraic expression involving x.