Problem 94 - Precalculus: Functions and Graphs

Question

Rewrite the expression in nonradical form without using absolute values for the indicated values of \(\theta .\) $$\sqrt{1+\cot ^{2} \theta}, \quad 0<\theta<\pi$$

Step-by-Step Solution

Answer

\(\csc \theta\) for \(0<\theta<\pi\).

Step 1: Recall Trigonometric Identity

We begin by recalling the Pythagorean identity for trigonometric functions: \(1 + an^2 \theta = \sec^2 \theta.\) However, we are interested in \(1 + \cot^2 \theta\), which is similar. We have: \[1 + \cot^2 \theta = \csc^2 \theta.\]

Step 2: Understand the Given Range for \(\theta\)

We are given that \(0 < \theta < \pi\). In this range, both \(\csc \theta\) and \(\cot \theta\) are defined. More importantly, \(\csc \theta\) is positive within this interval because \(\sin \theta\) is positive from \(0 < \theta < \pi/2\) and negative from \(\pi/2 < \theta < \pi\), making \(\csc \theta\) always positive.

Step 3: Apply the Identity to Simplify the Expression

Given the expression \(\sqrt{1 + \cot^2 \theta}\), substitute the identity we recalled: \(1 + \cot^2 \theta = \csc^2 \theta\). The expression becomes \[\sqrt{\csc^2 \theta}.\]

Step 4: Simplify \(\sqrt{\csc^2 \theta}\) Without Absolute Values

Normally, \(\sqrt{x^2} = |x|\). However, since \(\csc \theta\) is positive for \(0 < \theta < \pi\), we can simply have \(\sqrt{\csc^2 \theta} = \csc \theta\) without needing absolute values.

Key Concepts

Pythagorean Identity

The Pythagorean identities are crucial tools in trigonometry. They extend the famous Pythagorean theorem into the realm of trigonometric functions. For anyone exploring the trigonometric circle, these identities are like a compass. They provide direction and context.
One key identity you should remember is:

• \(\sin^2 \theta + \cos^2 \theta = 1\)

From this identity, we derive others, such as:

• \(1 + \tan^2 \theta = \sec^2 \theta\)
• \(1 + \cot^2 \theta = \csc^2 \theta\)

These relationships help us express squared trigonometric functions in terms of others. In the given exercise, we used the identity \(1 + \cot^2 \theta = \csc^2 \theta\) to manipulate the expression. This shows how the identities help in rewriting complex expressions, making them simpler to understand and solve.

Cosecant Function

The cosecant function, represented as \(\csc \theta\), is one of the reciprocal trigonometric functions. It is the reciprocal of the sine function. Simply put, \(\csc \theta = \frac{1}{\sin \theta}\). This means wherever the sine function is zero, the cosecant is undefined.
Understanding \(\csc \theta\) is crucial because it appears in various identities and formulas, such as \(1 + \cot^2 \theta = \csc^2 \theta\). In the context of the exercise, we determined \(\csc \theta\) is strictly positive between \(0 < \theta < \pi\). This feature is particularly important for simplifying expressions.

• In the interval \(0 < \theta < \pi/2\), both sine and cosecant are positive.
• In the interval \(\pi/2 < \theta < \pi\), sine is negative, but since cosecant is a reciprocal, it remains positive.

This consistent positivity allows us to remove absolute values, making the expression easier to simplify and comprehend.

Cotangent Function

The cotangent function is another reciprocal trigonometric function. It can be expressed as \(\cot \theta = \frac{1}{\tan \theta}\), or equivalently, \(\frac{\cos \theta}{\sin \theta}\). What's interesting about the cotangent function is its behavior and range. Understanding \(\cot \theta\) aids in solving a broad range of trigonometric problems.
In the interval \(0 < \theta < \pi\), \(\cot \theta\) behaves predictably:

• In \(0 < \theta < \pi/2\), both \(\cos \theta\) and \(\sin \theta\) are positive, so \(\cot \theta\) is positive.
• In \(\pi/2 < \theta < \pi\), \(\cos \theta\) is negative while \(\sin \theta\) is positive, making \(\cot \theta\) negative.

Being able to recognize these properties allows us to apply identities like \(1 + \cot^2 \theta = \csc^2 \theta\) effectively. Whenever we simplify expressions involving \(\cot \theta\), these insights make navigation through solutions clearer and more straightforward.