Question

Graph the following equations. Use a graphing utility to check your work and produce a final graph. $$r=\sin 3 \theta$$

Short answer

Q: Explain step by step how to graph a polar equation given by $$r = \sin{3\theta}$$. A: 1. Create a table with values of $$\theta$$, ranging from $$0$$ to $$2\pi$$ with increments of $$\pi/6$$ or $$\pi/12$$ (choose based on precision). 2. Plug each $$\theta$$ value into the equation $$r = \sin{3\theta}$$ and calculate the corresponding $$r$$ values. Add these to the table to create polar coordinate points $$(r, \theta)$$. 3. Plot the polar coordinate points $$(r, \theta)$$ on polar graph paper by rotating by the given angle $$\theta$$ and moving out from the origin by a distance of $$r$$. 4. Connect the points with a smooth curve to see the graph of the polar equation $$r = \sin{3\theta}$$. 5. Verify your work by comparing your hand-drawn graph with a graph produced by a graphing utility like Desmos or a graphing calculator. 6. If your hand-drawn graph matches the one produced by the graphing utility, your work is complete. If not, revisit the previous steps to identify any errors and make corrections as needed. Produce the final graph based on your verified work.

Step-by-step solution

Make a table with $$\theta$$ values

Create a table with values of $$\theta$$, ranging from $$0$$ to $$2\pi$$ with increments of $$\pi/6$$ or $$\pi/12$$ (choose based on precision).

Calculate the $$r$$ values for each $$\theta$$

Plug each $$\theta$$ value into the equation $$r = \sin{3\theta}$$. Calculate the corresponding $$r$$ values and add them to the table. This will give us polar coordinate points $$(r, \theta)$$.

Plot the points

Use the polar coordinate points $$(r, \theta)$$ from the table to plot the points on polar graph paper. To plot a polar coordinate point, start at the origin, rotate by the given angle $$\theta$$, and then move out from the origin by a distance of $$r$$.

Connect the points

With all the points plotted, we now have an idea of the shape the graph should take. Connect the points with a smooth curve to see the graph of the polar equation $$r = \sin{3\theta}$$.

Verify your work using a graphing utility

Using a graphing utility like Desmos or a graphing calculator, enter the polar equation $$r = \sin{3\theta}$$. Compare your hand-drawn graph with the one generated by the graphing utility to verify your work.

Produce the final graph

If your hand-drawn graph matches the one produced by the graphing utility, your work is complete. If not, revisit the previous steps to identify any errors and make corrections as needed. Produce the final graph based on your verified work.

Key concepts

Polar Equation

Polar equations are a type of mathematical expression that provides a way to represent curves on a plane using polar coordinates. Instead of using x and y coordinates, like in Cartesian coordinates, polar coordinates use a radius (r) and an angle (θ), hence the name polar.

Let's consider the polar equation from the exercise, \( r = \sin(3\theta) \). This equation describes a curve by stating how far from the origin (r) a point is, depending on the angle (θ) from the positive x-axis. As θ varies from \(0\) to \(2\pi\), r changes according to the sine function's behavior, but it's multiplied by three, which affects the curve's appearance on the graph. Variations of θ produce different r values, creating a set of points that, when plotted, will form a shape specific to the polar equation given.

Graphing Utility

Graphing utilities are incredibly helpful tools that can aid students in visualizing mathematical concepts, especially when dealing with complex functions like those in polar coordinates. These utilities can range from software applications, like Desmos, to advanced graphing calculators.

A graphing utility can graph the given polar equation \(r = \sin(3\theta)\) by computing r for a range of θ values and displaying the resulting curve. This visual check can verify the accuracy of a hand-drawn graph or help students comprehend the effects of altering parts of the equation. When using a graphing utility, you can often input the polar equation directly and adjust settings to ensure proper graphing parameters such as the scale for r and the range for θ.

Polar Graph Paper

Unlike Cartesian graph paper, which features a grid of squares, polar graph paper has concentric circles divided by radial lines that represent angles. Each circle represents a different radius from the origin, and the angles are usually marked in degrees or radians.

When graphing polar equations by hand, like \( r = \sin(3\theta) \), polar graph paper can be invaluable. To plot points, you find the angle on the paper corresponding to θ, then move outward to the circle with the radius equal to your r value. This specialized paper makes it easier to get an accurate representation of the polar curve, which could be more challenging to achieve on standard graph paper.

Trigonometric Functions

Trigonometric functions, including sine, cosine, and tangent, are fundamental in plotting polar equations. These functions relate the angles of a triangle to its sides, which in the context of polar coordinates, translates to the relationship between θ and r.

In the exercise's polar equation \(r = \sin(3\theta)\), the sine function determines the radius for each angle, and the multiplication by three leads to the curve having more 'petals' or loops than it would have if it were simply \(r = \sin(\theta)\). Understanding how trigonometric functions behave is crucial when graphing polar equations, as they define the curve's shape and the nature of its symmetry or periodicity.