Problem 39 - Precalculus with Calculus Previews

Question

Write the given expression as an algebraic expression in \(x\). $$ \csc (\arctan x) $$

Step-by-Step Solution

Answer

\( \csc(\arctan x) = \frac{\sqrt{x^2 + 1}}{x} \).

Step 1: Recall the Definitions

The cosecant function, \( \csc \theta \), is defined as \( \frac{1}{\sin \theta} \). The arctangent function, \( \arctan x \), is the angle whose tangent is \( x \). If we set \( \theta = \arctan x \), then \( \tan \theta = x \).

Step 2: Use a Right Triangle Representation

Imagine a right triangle where \( \theta \) is one angle and \( x = \frac{\text{opposite}}{\text{adjacent}} \). Set opposite side = \( x \) and adjacent side = 1. By Pythagoras' theorem, the hypotenuse = \( \sqrt{x^2 + 1} \).

Step 3: Express \( \sin \theta \)

In the right triangle, \( \sin \theta \) is the ratio of the opposite side to the hypotenuse. So, \( \sin \theta = \frac{x}{\sqrt{x^2 + 1}} \).

Step 4: Find \( \csc \theta \) Using \( \sin \theta \)

Recall that \( \csc \theta = \frac{1}{\sin \theta} \). So, \( \csc \theta = \frac{1}{\frac{x}{\sqrt{x^2 + 1}}} = \frac{\sqrt{x^2 + 1}}{x} \).

Step 5: Write the Final Algebraic Expression

Hence, \( \csc(\arctan x) = \frac{\sqrt{x^2 + 1}}{x} \).

Key Concepts

Arctangent Function

The arctangent function, often denoted as \( \arctan x \), is one of the inverse trigonometric functions. It is used to determine an angle whose tangent is a given number. In simpler terms, if you know the tangent of an angle, the arctangent can be used to find out what that angle actually is.

• Definition: If \( \theta = \arctan x \), then \( \tan \theta = x \).
• This implies that the angle \( \theta \) has a tangent of \( x \).
• Output range: For the arctangent function, the range is usually between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\) radians (or -90 to 90 degrees).

Since tangent represents the ratio of the opposite side to the adjacent side in a triangle, understanding the arctan function helps us recognize the relationship between a triangle's side lengths and angles.

Right Triangle Trigonometry

Right triangle trigonometry is a foundational tool for understanding relationships in a triangle where one angle is 90 degrees. It lends itself well to practical applications, such as solving trigonometric problems using side lengths and angles.

• We start by labeling the sides: opposite, adjacent, and hypotenuse.
• In our task, since \( \tan \theta = x \), we assume opposite equals \( x \) and adjacent equals 1.
• The hypotenuse, calculated using the Pythagorean theorem, becomes \( \sqrt{x^2 + 1} \).

Using the sine function for right triangles, \( \sin \theta \) is given by the ratio of the opposite to hypotenuse. Therefore, \( \sin \theta = \frac{x}{\sqrt{x^2 + 1}} \). Knowing \( \sin \theta \) helps us find \( \csc \theta \), as it is simply the reciprocal of \( \sin \theta \).

Pythagorean Theorem

The Pythagorean theorem is crucial in connecting the sides of right triangles. It's often expressed as \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides.
In our specific example where \( \tan \theta = x \), with opposite side \( x \) and adjacent side 1, the Pythagorean theorem allows us to calculate the length of the hypotenuse:

• Using the theorem: \( 1^2 + x^2 = (\text{hypotenuse})^2 \).
• Solving gives: hypotenuse = \( \sqrt{x^2 + 1} \).

This theorem aids in understanding the sine function, as knowing the hypotenuse helps calculate \( \sin \theta \), which is vital for finding \( \csc(\arctan x) \). The relationship between all sides in a right triangle is fundamental to expressing trigonometric functions algebraically.