Question

Use a graphing utility to (a) graph the polar equation, (b) draw the tangent line at the given value of \(\boldsymbol{\theta}\), and (c) find \(d y / d x\) at the given value of \(\theta\). (Hint: Let the increment between the values of \(\theta\) equal \(\pi / 24 .)\) $$ r=3 \sin \theta, \theta=\frac{\pi}{3} $$

Short answer

The tangent line at \(\theta=\frac{\pi}{3}\) can be represented in the Cartesian coordinate system as a function of x using the conversion process. The derivative \( \frac{dy}{dx}\) evaluated at \(\theta=\frac{\pi}{3}\) can be computed using the derivative formula for polar coordinates.

Step-by-step solution

Convert the equation to Cartesian coordinates

To represent a polar coordinate \(r=3 \sin \theta\) in Cartesian, use the conversion formulas \(x = r\cos \theta\) and \(y = r\sin \theta\). Substituting \(r = 3sin\theta\) into the equations, we get \(x = 3 \sin \theta \cos \theta\) and \(y = 3 \sin^2 \theta\).

Graph the function and draw the tangent line

You could do this with a graphing utility tool. Plot the graph of \(y = 3\sin^2\theta\) against \(x\). Now draw a line tangent to the curve at the point where \(\theta = \frac{\pi} {3}\). This line represents the slope of the curve at that point.

Calculate the derivative

To calculate \( \frac{dy}{dx}\), use the derivative formula for polar coordinates: \( \frac{dy}{dx} = \frac{dr/d\theta\sin(\theta)+r\cos(\theta)}{dr/d\theta\cos(\theta)-r\sin(\theta)} \). Now substitute \(r = 3\sin \theta\) and \(\theta = \frac{\pi} {3}\) into the formula and solve for \( \frac{dy}{dx}\).

Evaluate the derivative at the given theta value

Plug in the value of \(\theta = \frac{\pi} {3}\) into the \(\frac{dy}{dx}\) equation from step 3 and simplify. This will give you the value of the derivative at the given theta value.

Key concepts

Graphing Polar Coordinates

Understanding polar coordinates involves recognizing the way points are located on a plane using a radius and an angle as opposed to x and y values. To graph polar coordinates, remember that each point is represented as \( (r, \theta) \), where \( r \) is the radial distance from the origin (the length of the line segment from the origin to the point), and \( \theta \) is the angle formed by this radial line with the positive x-axis, measured in radians or degrees.

For example, to graph the polar equation \( r=3 \sin \theta \) for \( \theta=\frac{\pi}{3} \), you would find the radial distance by evaluating the sine function and then drawing a line at \( \theta = \frac{\pi}{3} \) extending from the origin the determined distance. Tools like graphing calculators or software can visualize this process by plotting multiple points as \( \theta \) varies and connecting them to form the graph of the polar equation.\

• Locate the origin which is the center of the polar coordinate system.
• Determine \( r \) by plugging the given \( \theta \) value into the polar equation.
• Draw the line at angle \( \theta \) and mark the point that is \( r \) units away from the origin.

By repeating this for multiple values of \( \theta \) and connecting these points, you can create a graphical representation of the polar equation.

Polar to Cartesian Conversion

When you encounter a polar equation, such as \( r=3 \sin \theta \), and want to express it in familiar Cartesian coordinates, you can use the conversion formulas \( x = r\cos \theta \) and \( y = r\sin \theta \). These formulas arise from the relationship between the polar and Cartesian coordinate systems. Applying these formulas translates the polar equation into the Cartesian plane, where \( x \) and \( y \) denote horizontal and vertical distances respectively.

Conversion Process

Using the given polar equation:

• First, express \( r \) in terms of \( \theta \) as given.
• Then, calculate \( x \) by multiplying \( r \) by \( \cos(\theta) \).
• Similarly, calculate \( y \) by multiplying \( r \) by \( \sin(\theta) \).

For the equation \( r=3 \sin \theta \) at \( \theta=\frac{\pi}{3} \), you'd plug in the value of \( \theta \) and evaluate to find \( x = 3 \sin(\frac{\pi}{3}) \cos(\frac{\pi}{3}) \) and \( y = 3 \sin^2(\frac{\pi}{3}) \). Converting gives you the Cartesian coordinates of the point on the graph associated with \( \theta=\frac{\pi}{3} \) in the polar equation.

Derivative of Polar Functions

The derivative of a polar function provides information about the slope of its graph at a given angle \( \theta \). For a polar equation like \( r=3 \sin \theta \), the derivative with respect to \( x \) is not as straightforward as in Cartesian coordinates because both \( r \) and \( \theta \) affect the horizontal position. Nevertheless, we can find the slope of the tangent line at a specific point using the formula:\[\frac{dy}{dx} = \frac{\frac{dr}{d\theta}\sin(\theta) + r\cos(\theta)}{\frac{dr}{d\theta}\cos(\theta) - r\sin(\theta)}\].

To derive \( \frac{dy}{dx} \) for our given polar equation at \( \theta=\frac{\pi}{3} \):

• First, find \( \frac{dr}{d\theta} \) by differentiating \( r \) with respect to \( \theta \) to get \( \frac{dr}{d\theta} = 3\cos(\theta) \).
• Then, substitute \( r \) and \( \frac{dr}{d\theta} \) into the formula.
• Simplify the resulting expression.
• Lastly, plug \( \theta=\frac{\pi}{3} \) into the simplified formula to get the value of \( \frac{dy}{dx} \) at the given \( \theta \).

This calculation will yield the slope of the tangent line to the polar curve at \( \theta=\frac{\pi}{3} \) - a crucial piece of information for understanding the behavior of the polar function at that point.