Problem 94 - Algebra and Trigonometry

Question

Without drawing a graph, describe the behavior of the basic cosine curve.

Step-by-Step Solution

Answer

A basic cosine function consists of a wave that oscillates between -1 and 1 over an interval of \(2\pi\). It starts at its peak of 1 at \(x = 0\), goes down to 0 at \(x = \pi/2\), hits its minimum at \(x = \pi\), comes back up to 0 at \(x = 3\pi/2\), and finishes its cycle back at its peak at \(x = 2\pi\). This pattern repeats every \(2\pi\) radians.

Step 1: Definition

A basic cosine function is given by \( y = \cos(x) \), where x is an input value, y is an output value and -1 ≤ y ≤ 1. The period of the function is between \( 0 \) and \( 2\pi \) and it repeats after every \( 2\pi \) radians.

Step 2: Function behavior

The function starts at a maximum of \( y = 1 \) when \( x = 0 \). It decreases to \( y = 0 \) when \(x = \pi/2 \), then continues to decrease to a minimum value of \( y = -1 \) at \( x = \pi \). The function then increases back to \( y = 0 \) when \(x = 3\pi/2 \) and finally back to the maximum \( y = 1 \) at \( x = 2\pi \).

Step 3: Function symmetry

Since the cosine function repeats every \( 2\pi \) radians and the same pattern of values is repeated, we can say that it displays both translational symmetry and even symmetry about the vertical line \( x = \pi \).