Problem 36 - Algebra and Trigonometry

Question

Rewrite the expression as an algebraic expression in x. $$ \cos \left(\tan ^{-1} x\right) $$

Step-by-Step Solution

Answer

\( \cos (\tan^{-1} x) = \frac{1}{\sqrt{x^2 + 1}} \).

Step 1: Understand the Problem

The expression involves the inverse tangent function, also known as arctan, and the cosine function. We want to find an algebraic form involving \( x \) for \( \cos (\tan^{-1} x) \).

Step 2: Use Trigonometric Identity

Recall the right triangle definitions and the identity \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \). Given that \( \tan^{-1} x = \theta \), we have \( \tan \theta = x = \frac{x}{1} \), implying the opposite side is \( x \) and the adjacent side is \( 1 \).

Step 3: Determine the Hypotenuse

Use the Pythagorean theorem to find the hypotenuse: \( \text{hypotenuse} = \sqrt{x^2 + 1^2} = \sqrt{x^2 + 1} \).

Step 4: Express \( \cos \theta \)

The cosine of an angle \( \theta \) in a right triangle is defined as \( \frac{\text{adjacent}}{\text{hypotenuse}} \). So, \( \cos \theta = \frac{1}{\sqrt{x^2 + 1}} \).

Step 5: Substitute Back

Since \( \theta = \tan^{-1} x \), we substitute back to get \( \cos (\tan^{-1} x) = \frac{1}{\sqrt{x^2 + 1}} \) as the algebraic expression in terms of \( x \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, also known as arc functions, are the opposites of the regular trigonometric functions. They allow you to find an angle when you know the value of its sine, cosine, or tangent. For example,

• arctan (also written as \( \tan^{-1} \)) gives you the angle whose tangent is a given number.

In our problem, \( \tan^{-1} x \) implies that we're looking for an angle \( \theta \) such that \( \tan \theta = x \). Remember, the range for arctan is

• between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \).

Inverse trigonometric functions are essential because they reverse the original function to allow us to solve triangles and model situations in various branches of mathematics.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that helps calculate the length of sides in a right triangle. It states:

• \( a^2 + b^2 = c^2 \)

where \( c \) is the hypotenuse (the side opposite the right angle), and \( a \) and \( b \) are the other two sides.
In our solution, we used the theorem to determine the hypotenuse of a right triangle where one angle \( \theta \) is \( \tan^{-1} x \). Given one side (adjacent) is \( 1 \) and the opposite side is \( x \), the hypotenuse is:

• \( \sqrt{x^2 + 1} \)

This is crucial for finding other trigonometric ratios like sine and cosine based on this angle.

Right Triangle

A right triangle is a type of triangle that has one angle equal to \( 90^\circ \). It is exceptionally useful in trigonometry because it sets the stage for defining sine, cosine, and tangent in terms of its sides.
In a right triangle:

• The "opposite" side is the one opposite to the angle you're considering.
• The "adjacent" side lies next to the angle and forms the right angle with the opposite side.
• The "hypotenuse" is the longest side, always opposite the right angle.

In the case of \( \tan^{-1} x \), which gives you an angle \( \theta \), knowing that \( \tan \theta = x \) means you can represent \( x \) as \( \frac{x}{1} \), with the opposite being \( x \) and the adjacent being \( 1 \). Utilizing this right triangle framework, we calculate values like \( \cos \theta \), highlighting the power of right triangles in solving trigonometric problems.