Given: 5cos6Θ
What is the shape of the function?
A. Limacon
B. Rose
C. Lemniscate
D. Circle
if n is even,
r = a cos(n?) is a rosette with 2n leaves.
Think about it. cos(n?) goes from 1 to 0 every time n? grows by pi/2. So, in [0,2pi], 6? goes in and out 12 times.
You can always verify your work online, as at
http://www.wolframalpha.com/input/?i=r%3D5cos(6%CE%B8)
To determine the shape of the function, we need to analyze the given equation: 5cos(6Θ).
The shape of the function is determined by the parent function, cosine (cos), and any transformations applied to it. In this case, the amplitude of the cosine function is 5, which means the graph will oscillate between -5 and 5.
The argument of the cosine function is 6Θ. When the argument of the cosine function is Θ (without any coefficients), it results in a full period of oscillation (cycle) between 0 and 2π radians (360 degrees). However, when a coefficient is applied to the argument, the period is affected. In this case, the coefficient is 6.
To find the new period, we divide the original period (2π) by the coefficient: 2π / 6 = π/3. So, the graph will complete one full cycle (period) every π/3 radians.
Based on the given equation, 5cos(6Θ), the function will oscillate between -5 and 5 with a period of π/3 radians. This shape is known as a limaçon, which is a type of cardioid (heart-shaped curve) with a loop.
Therefore, the correct answer is A. Limacon.