Problem 27 - Algebra and Trigonometry

Question

Test for symmetry and then graph each polar equation. $$r=4 \sin 3 \theta$$

Step-by-Step Solution

Answer

The given equation is symmetric with respect to the origin. It will result in a rose with three petals, each of length 4 units.

Step 1: Symmetry With Respect To The Polar Axis

Substitute '(-θ)' for 'θ' in the given equation and simplify. Given, \(r = 4\sin(3\theta)\).Let's substitute \(-\theta\) for \(\theta\).If the result is equivalent to the original equation, then the graph is symmetric with respect to the polar axis. Here, \(r = 4\sin(3(-\theta)) = 4\sin(-3\theta)\). Since \(\sin(-\theta) = - \sin(\theta)\), the equation simplifies to \(r = -4\sin(3\theta)\) which is not equivalent to the original equation, so the graph is not symmetric about the polar axis.

Step 2: Symmetry With Respect To The Pole

Substitute '(-r)' for 'r' in the given equation and simplify.So, substitute \(-r = 4\sin(3\theta)\) for \(r\). If the result is equivalent to the original equation, then the graph is symmetric with respect to the pole. But in this case, \(-r = 4\sin(3\theta)\) is not equivalent to the original equation, which means the graph is not symmetric about the pole.

Step 3: Symmetry With Respect To The Origin

Substitute '(-θ)' for 'θ' and '(-r)' for 'r' in the given equation.After substitution, we get \(-r = 4\sin(3(-\theta)) = -4\sin(3\theta)\). But \(\sin(-\theta) = - \sin(\theta)\). Therefore, the equation becomes \(-r = -4\sin(3\theta)\) or \(r = 4\sin(3\theta)\), which is the same as the original equation. So, the graph is symmetric about the origin.

Step 4: Graph the Polar Equation

To graph this equation, select values for \(\theta\) between 0 and \(2\pi\) and then find corresponding 'r' values. Plot these points. The equation will create a rose with three leaves because of the '3' in \(\sin(3\theta)\), which indicates the number of leaves on the rose. The '4' in 'r=4sin3θ' represents the length of the petals.