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.sin (tan⁻¹ x)

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1. Consider a right triangle where \( \theta = \tan^{-1}(x) \). This means that \( \tan(\theta) = x \).
2. Since \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \), we can set the opposite side to \( x \) and the adjacent side to \( 1 \).
3. Use the Pythagorean theorem to find the hypotenuse: \( \text{hypotenuse} = \sqrt{x^2 + 1^2} = \sqrt{x^2 + 1} \).
4. Now, find \( \sin(\theta) \) which is \( \frac{\text{opposite}}{\text{hypotenuse}} = \frac{x}{\sqrt{x^2 + 1}} \).
5. Thus, \( \sin(\tan^{-1}(x)) = \frac{x}{\sqrt{x^2 + 1}} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as tan⁻¹ (arctangent), are used to find angles when the value of a trigonometric function is known. For example, if tan(θ) = x, then θ = tan⁻¹(x). Understanding how to interpret these functions is crucial for converting between angles and their corresponding trigonometric ratios.

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 tan(θ) = opposite/adjacent. These relationships allow us to express trigonometric functions in terms of the sides of a triangle, which is essential for solving problems involving inverse functions.

Pythagorean Identity

The Pythagorean identity states that for any angle θ, sin²(θ) + cos²(θ) = 1. This identity is fundamental in trigonometry as it connects the sine and cosine functions. When working with inverse trigonometric functions, this identity can be used to derive relationships between the sides of a right triangle, facilitating the conversion of expressions like sin(tan⁻¹(x)) into algebraic forms.