Problem 101 - A Graphical Approach to Precalculus with Limits

Question

Write each expression as an algebraic expression in \(u, u>0\). $$\sec \left(\cos ^{-1} u\right)$$

Step-by-Step Solution

Answer

\(\sec \left(\cos^{-1} u\right) = \frac{1}{u} , u > 0\).

Step 1: Understand the Inverse Function

The function \( \cos^{-1} u \) gives the angle \( \theta \) such that \( \cos \theta = u \). Thus, \( \theta = \cos^{-1} u \). Our goal is to express the secant of \( \theta \) using \( u \).

Step 2: Recall the Definition of Secant

Secant is the reciprocal of cosine. So, \( \sec(\theta) = \frac{1}{\cos(\theta)} \). Since we know that \( \cos(\theta) = u \), it follows that \( \sec(\theta) = \frac{1}{u} \).

Step 3: Write in Terms of \(u\)

Substitute \( \theta = \cos^{-1} u \) into the expression for secant: \( \sec(\cos^{-1} u) = \frac{1}{u} \). This expresses the secant in terms of \( u \), assuming \( u > 0 \).

Key Concepts

Inverse Functions

Inverse functions are a fascinating concept in mathematics. They allow us to reverse the roles of inputs and outputs in a function. In other words, if a function takes an input and produces an output, its inverse function will take that output and return to the original input.

For trigonometric functions, inverse functions help us find angles when given a ratio. For example:

• If you know the cosine of an angle and want to determine the angle itself, you can use the inverse cosine function, denoted as \( \cos^{-1} \).
• The expression \( \cos^{-1}(u) \) represents the angle \( \theta \) such that \( \cos(\theta) = u \).

Inverse trigonometric functions are essential in solving problems involving angles in various fields, such as physics and engineering. They allow for a systematic approach to solving equations where the unknown quantity is an angle.

Secant

The secant function is one of the lesser-known trigonometric functions, but it is very useful in higher-level mathematics. It is defined as the reciprocal of the cosine function. Mathematically, it can be expressed as:

• \( \sec(\theta) = \frac{1}{\cos(\theta)} \)
• Using this definition, if \( \cos(\theta) = u \), then \( \sec(\theta) = \frac{1}{u} \).

This relationship highlights how secant goes beyond just being the inverse of cosine by playing a pivotal role in relating angles and their cosine values in geometry and trigonometry.

Secant is particularly significant in calculus and mathematical analysis where various integrals and derivatives involve reciprocal trigonometric functions. Understanding this concept is crucial for handling complex problems with ease. It also provides an insight into how all trigonometric functions interconnect and relate through their definitions.

Cosine

Cosine is one of the primary trigonometric functions. It relates to the ratio of the adjacent side to the hypotenuse in a right-angled triangle. Its relationship with angles plays a critical role in various mathematical problems.

• Cosine can be written as \( \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \).
• It describes oscillatory behaviors, such as waves seen in physics.

The cosine function is periodic and symmetric, meaning it repeats itself after a certain interval, known as the period. It is part of the fundamental sine and cosine waves that form the basis of Fourier analysis and other advanced mathematical techniques.

In our problem, the cosine function's inverse helped determine the angle \( \theta \) for which the cosine is equal to \( u \). Knowing cosine values can quickly allow us to find related trigonometric functions such as secant. This interconnectedness is key in solving trigonometry problems efficiently.