Question

Plot the curve, \(\quad r=a \cos 3 \theta\), taking \(a=5 \mathrm{~cm} .\) Show that $$ \cot \psi=-3 \tan 3 \theta. $$

Short answer

To summarize, we converted the polar equation \(r = a \cos 3\theta\) into Cartesian coordinates to plot the curve using a graphing tool. After plotting the curve, we derived the relationship between \(\cot \psi\) and \(\tan 3\theta\) by utilizing polar and Cartesian coordinates and trigonometric identities. In the end, we showed that \(\cot \psi = -3 \tan 3\theta\).

Step-by-step solution

Convert the polar equation to Cartesian coordinates

Using the conversion formulas \(x=r\cos\theta\) and \(y=r\sin\theta\), we first find the values of \(r\cos\theta\) and \(r\sin\theta\) with \(r = 5\cos 3\theta\): $$ x = r\cos\theta = (5\cos 3\theta)\cos\theta \\ y = r\sin\theta = (5\cos 3\theta)\sin\theta $$

Plot the curve using a graphing tool

With these two equations, we can input them into a graphing tool such as Desmos, Wolfram Alpha, or any other preferred platform. By doing so, we are able to observe the relationship between \(x\) and \(y\) as we trace through the curve. The shape of the curve is called a rose curve of three petals, with a length of 5 cm for each petal.

Derive the relationship between \(\cot \psi\) and \(\tan 3\theta\)

We will first use the relation between polar and Cartesian coordinates: \(\tan\psi = \frac{y}{x}\). Taking the reciprocal of both sides, we get \(\cot\psi=\frac{x}{y}\). Plugging in the expressions for \(x\) and \(y\) from Step 1, we obtain: $$ \cot\psi=\frac{5\cos 3\theta\cos\theta}{5\cos 3\theta\sin\theta} $$ We can now simplify the expression: $$ \cot\psi=-\frac{\cos 3\theta\cos\theta}{\cos 3\theta\sin\theta}=-\frac{\cos\theta}{\sin\theta} $$ Now we apply the trigonometric identity \(\cos\theta=\frac{1}{\cot\theta}\) and the reciprocal of the double angle formula for tangent, \(\tan 2\theta=\frac{1}{\cot\theta-\cot\theta}\), we get: $$ \cot\psi=-\frac{1/\cot\theta}{\sin\theta}=-\frac{1}{(\cot\theta)(\sin\theta)}=-\frac{1}{(\cot\theta)((\cot\theta-\cot\theta\tan 2\theta))} $$ Simplifying further: $$ \cot\psi=-\frac{1}{(1-\cot\theta\tan 2\theta)} $$ Now, we know that \(\tan 2\theta=3\tan\theta\) and \(\cot\theta=\frac{1}{\tan\theta}\), so: $$ \cot\psi=-\frac{1}{(1-(3\tan\theta)(\cot\theta))}=-\frac{1}{(1-3\tan\theta)(\frac{1}{\tan\theta})} $$ Finally, we get the desired result: $$ \cot\psi=-3\tan 3\theta $$ We have now demonstrated that \(\cot \psi = -3 \tan 3\theta\).

Key concepts

Polar to Cartesian Conversion

Understanding how to convert polar coordinates to Cartesian coordinates is essential in calculus, especially when graphing curves that are naturally described in polar terms. Polar coordinates consist of a radial distance, often denoted as 'r', and an angle 'θ', whereas Cartesian coordinates are in the form of (x, y) representing points on a horizontal and vertical axis respectively.

To convert a point from polar to Cartesian coordinates, we use the following relationships: \(x = r\cos(θ)\) and \(y = r\sin(θ)\). These formulas emanate directly from the definitions of the sine and cosine functions in a right-angled triangle. By substitution, a polar equation can be rewritten as a set of parametric equations for x and y, providing a bridge to the Cartesian plane.

For example, given the polar equation \(r = 5\cos(3θ)\), we convert to Cartesian by multiplying r with the cosine and sine of θ, leading us to \(x = (5\cos(3θ))\cos(θ)\) and \(y = (5\cos(3θ))\sin(θ)\). This step is crucial when graphing the curve or creating a visualization using graphing tools that typically operate in Cartesian coordinates.

Rose Curve Graphing

Rose curves are a fascinating family of sinusoidal curves that often appear in polar coordinate graphing. They look like petals of a rose, hence their name. The general form of a rose curve is given by \(r = a\cos(kθ)\) or \(r = a\sin(kθ)\), where 'a' is the amplitude that dictates the length of the petals, and 'k' determines the number of petals.

When graphing a rose curve, such as \(r = 5\cos(3θ)\), you'll notice that the value of 'k' is 3, indicating that the curve will have 3 petals if k is odd, and 2k petals if k is even. The petals extend from the origin out to the distance 'a', so each of our petals will have a length of 5 cm since that is the value of 'a'.

Drawing a Rose Curve Step by Step:

• Plot points on the polar grid by calculating r for various values of θ.
• Notice the curve's symmetry to simplify the plotting process.
• Join the points to form the petals, ensuring they are smooth and rounded.

Graphing technology can streamline this process, but it's educational to plot a few points by hand to truly understand the geometry of rose curves.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables where both sides of the identity are defined. These identities are not only useful in simplifying expressions and solving equations but also in converting between different trigonometric functions.

One of the basic identities is the Pythagorean identity, which states that \(\sin^2(θ) + \cos^2(θ) = 1\). From there, other reciprocal identities such as \(\cot(θ) = \frac{1}{\tan(θ)}\) follow. And the double-angle formulas, such as \(\tan(2θ) = \frac{2\tan(θ)}{1 - \tan^2(θ)}\), are vital when dealing with expressions involving trigonometric functions of multiple angles, like in our example with \(\cot(ψ)\) and \(\tan(3θ)\).

Understanding these identities allows for the manipulation and simplification of equations as seen in the exercise, where we use various trigonometric identities to show that \(\cot(ψ) = -3\tan(3θ)\). Breaking down complex trigonometric expressions often involves strategically applying these identities to reveal simpler forms or relationships between angles and their trigonometric ratios.