Problem 25 - University Calculus: Early Transcendentals

Question

Find \(d r / d \theta\). $$r=\sec \theta \csc \theta$$

Step-by-Step Solution

Answer

\(\frac{d r}{d\theta} = -4 \cot(2\theta) \csc(2\theta)\).

Step 1: Understand the expression

First, let's understand the given expression for the radius function in polar coordinates: \(r = \sec \theta \csc \theta\). Here, \(\sec \theta\) is the secant of \(\theta\) which is \(\frac{1}{\cos \theta}\), and \(\csc \theta\) is the cosecant of \(\theta\) which is \(\frac{1}{\sin \theta}\). So the expression can be rewritten as: \[r = \frac{1}{\cos \theta} \cdot \frac{1}{\sin \theta} = \frac{1}{\cos \theta \cdot \sin \theta} = \frac{1}{\sin(2\theta)/2}\] And further simplification using the double angle formula gives us: \[r = \frac{2}{\sin(2\theta)}\]

Step 2: Differentiate the expression

Now, differentiate the expression \(r = \frac{2}{\sin(2\theta)}\) with respect to \(\theta\). Using the quotient rule, we have: \[\frac{d}{d\theta} \left( \frac{2}{\sin(2\theta)} \right) = \frac{0 \cdot \sin(2\theta) - 2 \cdot \cos(2\theta) \cdot 2}{(\sin(2\theta))^2}\] Which simplifies to: \[\frac{d r}{d\theta} = \frac{-4 \cos(2\theta)}{\sin^2(2\theta)}\] This is the derivative of \(r\) with respect to \(\theta\).

Step 3: Simplify the derived expression

The derivative \(\frac{d r}{d\theta} = \frac{-4 \cos(2\theta)}{\sin^2(2\theta)}\) can be further simplified. Knowing that \(\cot(x) = \frac{\cos(x)}{\sin(x)}\), we can rewrite the derivative as: \[\frac{d r}{d\theta} = -4 \frac{\cos(2\theta)}{\sin^2(2\theta)} = -4 \cot(2\theta) \cdot \csc(2\theta)\] Thus, \(\frac{d r}{d\theta} = -4 \cot(2\theta) \csc(2\theta)\) is the most simplified form of the derivative.

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in mathematics that simplify expressions and solve equations involving trigonometric functions. In this exercise, we begin by using identities such as secant (\(\sec \theta = \frac{1}{\cos \theta}\)) and cosecant (\(\csc \theta = \frac{1}{\sin \theta}\)) to break down the expression for the radius function (\(r = \sec \theta \csc \theta\)).
When rewritten using these identities, we have:

• \(r = \frac{1}{\cos \theta \cdot \sin \theta}\)
• \(r = \frac{1}{\sin(2\theta)/2}\), using the double angle identity for sine
• \(r = \frac{2}{\sin(2\theta)}\)

Understanding how to apply these identities allows us to transform complex trigonometric functions into more manageable forms, laying the groundwork for finding derivatives.

Quotient Rule

The Quotient Rule is crucial when differentiating functions that are expressed as a fraction of two functions. This rule states:\[ \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2}\]where \(u\) and \(v\) are differentiable functions of \(x\).
In the problem, to differentiate \(\frac{2}{\sin(2\theta)}\) with respect to \(\theta\), we know:

• \(u = 2\) and \(du/d\theta = 0\)
• \(v = \sin(2\theta)\) and \(dv/d\theta = 2\cos(2\theta)\)

Applying the quotient rule gives:\[ \frac{d}{d\theta} \left( \frac{2}{\sin(2\theta)} \right) = \frac{0 \cdot \sin(2\theta) - 2 \cdot 2\cos(2\theta)}{(\sin(2\theta))^2} \]which simplifies to \(\frac{d r}{d\theta} = \frac{-4 \cos(2\theta)}{\sin^2(2\theta)}\).
Understanding the quotient rule enables us to systematically handle fractions in calculus.

Differentiation Simplification

Differentiation simplification refers to making a derived expression easier to interpret and work with. After applying the quotient rule, we get:\[ \frac{-4 \cos(2\theta)}{\sin^2(2\theta)} \]
This can be further simplified using the trigonometric identity \(\cot(x) = \frac{\cos(x)}{\sin(x)}\). By rewriting this part of the derivative, we can express it as:

• \(-4 \cdot \frac{\cos(2\theta)}{\sin^2(2\theta)} = -4 \cot(2\theta) \cdot \csc(2\theta)\)

The transformation to use cotangent (\(\cot\)) and cosecant (\(\csc\)) not only reduces the complexity but also highlights the relationships between the trigonometric functions.
This process of simplification is pivotal in calculus to ensure results are in their most elegant and usable form.