(Use of Tech) Finger curves: r = f(θ) = cos(aᶿ) - 1.5, where a = (1 + 12π)^(1/(2π)) ≈ 1.78933
a. Show that f(0) = f(2π) and find the point on the curve that corresponds to θ = 0 and θ = 2π.

Verified step by step guidance

1. First, write down the given function explicitly: \(r = f(\theta) = \cos(a\theta) - 1.5\), where \(a = (1 + 12\pi)^{\frac{1}{2\pi}}\).
2. To show that \(f(0) = f(2\pi)\), substitute \(\theta = 0\) into the function: \(f(0) = \cos(a \cdot 0) - 1.5 = \cos(0) - 1.5\).
3. Next, substitute \(\theta = 2\pi\) into the function: \(f(2\pi) = \cos(a \cdot 2\pi) - 1.5\).
4. Use the property of the cosine function that \(\cos(\alpha + 2\pi k) = \cos(\alpha)\) for any integer \(k\). Since \(a \cdot 2\pi\) can be expressed in terms of an integer plus a fractional part, analyze \(a \cdot 2\pi\) to see if it corresponds to an integer multiple of \$2\pi$ or how it affects the cosine value.
5. Finally, find the points on the curve for \(\theta = 0\) and \(\theta = 2\pi\) by converting from polar coordinates \((r, \theta)\) to Cartesian coordinates using \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\) for each value of \(\theta\).

Key Concepts

Polar Coordinates and Parametric Curves

Polar coordinates represent points using a radius r and angle θ, where r = f(θ) defines a curve. Understanding how to interpret and plot points given by r and θ is essential, especially when θ varies over an interval like [0, 2π]. This helps in analyzing the shape and properties of curves defined in polar form.

Periodicity and Function Evaluation

Periodicity refers to a function repeating its values at regular intervals. Showing that f(0) = f(2π) involves evaluating the function at these points and understanding the behavior of trigonometric functions like cosine, which are periodic with period 2π. This concept is key to proving equality and identifying points on the curve.

Substitution and Simplification of Expressions

Substitution involves replacing variables with given values to evaluate functions. Simplifying expressions, especially those involving exponents and trigonometric functions, is necessary to find exact values of r at specific θ values. This skill is crucial for determining the coordinates of points on the curve.