Properties of Cosine Functions - Algebra-2

1. Fundamental Concepts

Definition: The cosine function, denoted as $$\cos(x)$$ , is a periodic function with a period of $$2\pi$$ . It represents the x-coordinate of a point on the unit circle corresponding to an angle $$x$$ .
Domain and Range: The domain of $$\cos(x)$$ is all real numbers, $$(-\infty, \infty)$$ , and its range is $$[-1, 1]$$ .
Symmetry: The cosine function is even, meaning $$\cos(-x) = \cos(x)$$ for all $$x$$ in its domain.

2. Key Concepts

Period: Repeats every $2\pi$ radians (period = $2\pi$ ), so $\cos(x + 2\pi) = \cos x$ for any real $x$ .

Amplitude: The maximum distance from the midline (y = 0) to the peak/trough, equal to $|1| = 1$ (no vertical stretch/compression in the standard function).

Symmetry: Even function, satisfying $\cos(-x) = \cos x$ (symmetric about the y-axis).

Key Points: Critical points in one period ( $0$ to $2\pi$ ):

$(0, 1)$ (starting point, maximum)

$(\frac{\pi}{2}, 0)$ (midline)

$(\pi, -1)$ (minimum, trough)

$(\frac{3\pi}{2}, 0)$ (midline)

$(2\pi, 1)$ (end of one period, maximum)

3. Examples

Easy

Find the values of $\cos(0)$ and $\cos(\pi)$ .

Solution: Use key points of the cosine function:

$\cos(0) = 1$ (maximum), $\cos(\pi) = -1$ (minimum).

Medium

Verify if $\cos(-\frac{\pi}{2}) = \cos(\frac{3\pi}{2})$ .

Solution:

1. Use even function property: $\cos(-\frac{\pi}{2}) = \cos(\frac{\pi}{2}) = 0$ .

2. From key points: $\cos(\frac{3\pi}{2}) = 0$ .

3. Thus, $\cos(-\frac{\pi}{2}) = \cos(\frac{3\pi}{2})$ (true).

Hard

For $y = \cos x$ , find all $x$ in $[0, 3\pi]$ where $\cos x = 0$ .

Solution:

1. In one period ( $0, 2\pi$ ), $\cos x = 0$ at $x = \frac{\pi}{2}$ and $x = \frac{3\pi}{2}$ (key points).

2. Extend to $[2\pi, 3\pi]$ : add the period ( $2\pi$ ) to $\frac{\pi}{2}$ , getting $x = 2\pi + \frac{\pi}{2} = \frac{5\pi}{2}$ .

3. Final solutions: $x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}$ .

4. Problem-Solving Techniques

Reference Unit Circle: Always refer to the unit circle for standard values and symmetries.
Use Identities: Apply trigonometric identities such as $$\cos^2(x) + \sin^2(x) = 1$$ to solve complex equations.
Graphical Interpretation: Use graphs to visualize solutions and understand periodicity and symmetry.