Problem 23 - Precalculus: Functions and Graphs

Question

Write the expression as an algebraic expression in \(x\) for \(x>0\). $$\sin \left(\tan ^{-1} x\right)$$

Step-by-Step Solution

Answer

\( \frac{x}{\sqrt{x^2 + 1}} \)

Step 1: Understanding the Relationship

First, let's consider the inverse tangent function, \( \tan^{-1}(x) \), which is also known as arctan. This function returns the angle \( \theta \) such that \( \tan(\theta) = x \).

Step 2: Setting Up the Right Triangle

Visualize this situation as a right triangle where \( \theta \) is one of the angles. Recall that \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \). In this scenario, \( \tan(\theta) = x = \frac{x}{1} \), so the opposite side is \( x \) and the adjacent side is \( 1 \).

Step 3: Finding the Hypotenuse

To find the hypotenuse of the triangle, use the Pythagorean Theorem: \( c = \sqrt{x^2 + 1^2} = \sqrt{x^2 + 1} \). Thus, the hypotenuse of the triangle is \( \sqrt{x^2 + 1} \).

Step 4: Expressing \(\sin(\theta)\) in Terms of \(x\)

The sine function is given by \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \). So, \( \sin(\tan^{-1}(x)) = \frac{x}{\sqrt{x^2 + 1}} \). This is the algebraic expression in \( x \) that we were asked to find.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, like \( \tan^{-1}(x) \), are crucial in connecting geometric angles to specific ratios. They take a ratio and return the corresponding angle. For example, \( \tan^{-1}(x) \) gives the angle \( \theta \) whose tangent is \( x \). These functions are often called arctangent, arcsine, and arccosine in various contexts.

• Arctangent: \( \tan^{-1}(x) \) relates to the angle of a right triangle, such that \( \tan(\theta) = x \).
• Range: Inverse trigonometric functions have restricted ranges to be defined properly. For \( \tan^{-1}(x) \), the range is \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \).
• Application: These functions are vital in calculus and analytical geometry, assisting in integration, differentiation, and solving equations involving angles.

Understanding inverse trigonometric functions not only helps in trigonometry problems but also enhances the grasp of mathematical modeling involving angles and real-world situations.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that connects the sides of a right-angled triangle. It is expressed as \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse, the longest side of the triangle, and \( a \) and \( b \) are the other two sides.

• Hypotenuse Calculation: In the context of \( \tan^{-1}(x) \), if we form a right triangle with legs \( x \) and \( 1 \), the hypotenuse becomes \( \sqrt{x^2 + 1} \) through this theorem.
• Applications: Used in various fields like physics, engineering, and architecture, it forms the basis for trigonometric identities and relationships.

This theorem is not only pivotal to solving geometric problems but also underpins key trigonometric identities and transformations in advanced mathematics. It aids in visualizing and computing relationships in right triangles and understanding spatial dimensions.

Sine Function

The sine function, noted as \( \sin(\theta) \), is one primary trigonometric function used to relate an angle of a right triangle to the lengths of its sides. Specifically, it is defined as the ratio of the length of the side opposite the angle to the hypotenuse.

• Basic Formula: \( \sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}} \).
• In Context: In the example, \( \sin(\tan^{-1}(x)) = \frac{x}{\sqrt{x^2 + 1}} \), derived from the constructed triangle with opposite \( x \) and hypotenuse \( \sqrt{x^2 + 1} \).
• Sine Wave: The sine function is also known for creating periodic sine waves, which are essential in studying waves, oscillations, and electrical circuits.

Understanding sine is crucial for comprehending wave mechanics, harmonic motion in physics, and solving problems involving periodic phenomena.