Problem 242 - Calculus Volume 3

Question

Find the slope of a tangent line to a polar curve \(r=f(\theta)\). Let \(x=r \cos \theta=f(\theta) \cos \theta\) and \(y=r \sin \theta=f(\theta) \sin \theta, \quad\) so the polar equation \(r=f(\theta)\) is now written in parametric form. $$ r=2 \sin (3 \theta) ; \text { tips of the leaves } $$

Step-by-Step Solution

Answer

The slope of the tangent line is given by \( \frac{3 \cos(3\theta) \sin\theta + \sin(3\theta) \cos\theta}{3 \cos(3\theta) \cos\theta - \sin(3\theta) \sin\theta} \).

Step 1: Understand the Problem

The given polar curve equation is \( r = 2 \sin(3\theta) \). We need to find the slope of the tangent line at any point on the curve. The transformation of polar to parametric equations allows calculating the slope of the tangent using derivatives.

Step 2: Convert to Parametric Equations

Convert the polar coordinates into parametric equations using \( x = r \cos \theta \) and \( y = r \sin \theta \). For \( r = 2 \sin(3\theta) \), we have:\[ x = 2 \sin(3\theta) \cos \theta \] \[ y = 2 \sin(3\theta) \sin \theta \]

Step 3: Find Derivatives

To find the slope of the tangent line, calculate \( \frac{dx}{d\theta} \) and \( \frac{dy}{d\theta} \):\( dx/d\theta = \frac{d}{d\theta} \left( 2 \sin(3\theta) \cos\theta \right) \)Apply product rule to differentiate: \[ \frac{dx}{d\theta} = 2 \left( 3 \cos(3\theta) \cos\theta - \sin(3\theta) \sin\theta \right) \]Similarly, for \( y \), differentiate:\( dy/d\theta = \frac{d}{d\theta} \left( 2 \sin(3\theta) \sin\theta \right) \)Apply product rule:\[ \frac{dy}{d\theta} = 2 \left( 3 \cos(3\theta) \sin\theta + \sin(3\theta) \cos\theta \right) \]

Step 4: Calculate Slope of Tangent Line

The slope of the tangent line in parametric form is given by \( \frac{dy/d\theta}{dx/d\theta} \). Substitute the derivatives from Step 3:\[ \text{slope} = \frac{2 \left( 3 \cos(3\theta) \sin\theta + \sin(3\theta) \cos\theta \right)}{2 \left( 3 \cos(3\theta) \cos\theta - \sin(3\theta) \sin\theta \right)} \]Simplify:\[ \text{slope} = \frac{3 \cos(3\theta) \sin\theta + \sin(3\theta) \cos\theta}{3 \cos(3\theta) \cos\theta - \sin(3\theta) \sin\theta} \]

Step 5: Simplify the Expression

Utilize trigonometric identities to simplify the slope expression.\[ \text{slope} = \frac{3 \sin(\theta + 3\theta) + \sin(\theta - 3\theta)}{3 \cos(\theta + 3\theta) - \cos(\theta - 3\theta)} \]This expression represents the slope of the tangent line at any \( \theta \) on this polar curve.

Key Concepts

Parametric Equations

Parametric equations provide a powerful way to express a curve in a plane using one or more independent variables. In the context of polar coordinates, these independent variables are often based on an angle \(\theta\). By converting polar equations into parametric form, we can more easily apply calculus techniques to solve problems such as finding tangent lines and slopes.

In polar coordinates, a curve is defined as \(r = f(\theta)\). To convert this into parametric form, we use the relationships:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)

This conversion is particularly useful because it allows us to use derivatives with respect to \(\theta\), facilitating the computation of slopes of tangent lines and other calculus-based analyses.
For the problem at hand, the polar equation \(r = 2 \sin(3\theta)\) is converted into parametric equations:

• \(x = 2 \sin(3\theta) \cos \theta\)
• \(y = 2 \sin(3\theta) \sin \theta\)

Tangent Line Slope

Finding the slope of a tangent line to a curve in parametric form involves differentiating the expressions for \(x\) and \(y\) with respect to their parameter \(\theta\). The slope of the tangent line at any point on the curve is given by the ratio \(\frac{dy/d\theta}{dx/d\theta}\). Calculating these derivatives allows us to determine how the curve changes direction at each point.

The derivatives with respect to \(\theta\) are:

• \(\frac{dx}{d\theta} = 2 (3 \cos(3\theta) \cos \theta - \sin(3\theta) \sin \theta)\)
• \(\frac{dy}{d\theta} = 2 (3 \cos(3\theta) \sin \theta + \sin(3\theta) \cos \theta)\)

By substituting these derivatives into the slope formula, we find:

• \(\text{slope} = \frac{3 \cos(3\theta) \sin \theta + \sin(3\theta) \cos \theta}{3 \cos(3\theta) \cos \theta - \sin(3\theta) \sin \theta}\)

This provides the slope at any point \(\theta\) on the polar curve. Understanding this process is crucial for analyzing the geometry of curves expressed in parametric forms.

Trigonometric Identities

Trigonometric identities play an essential role in simplifying mathematical expressions, especially when dealing with derivatives and slopes in polar and parametric equations. These identities help us transform and simplify expressions to reach more understandable forms.

For example, in the expression of the slope of the tangent line, using identities like the angle addition and subtraction formulas helps simplify the complex trigonometric components. The term \(3 \sin(\theta + 3\theta) + \sin(\theta - 3\theta)\) is simplified using angle sum identities:

• \(\sin(a + b) = \sin a \cos b + \cos a \sin b\)
• \(\cos(a + b) = \cos a \cos b - \sin a \sin b\)

These identities help reduce the complexity of these expressions, offering clearer insights into the problems. In the case of the tangent slope equation, they help provide a concise representation of the slope calculation, effectively utilizing trigonometric simplifications for a cleaner solution.