AOS1 Topic 6: Circular Functions (COS)

Circular functions, also known as trigonometric functions, are fundamental in the study of angles and periodic phenomena. One of the most important circular functions is the cosine function, abbreviated as cos. It is widely used in mathematics, physics, engineering, and other fields to model periodic behavior such as waves, oscillations, and circular motion.

Definition of the Cosine Function

The cosine of an angle θ in a right-angled triangle is defined as the ratio of the adjacent side to the hypotenuse:

cos(θ) = Adjacent / Hypotenuse

For an angle θ in the unit circle, where the radius is 1, the cosine of θ represents the horizontal coordinate of the point on the unit circle corresponding to θ. This geometric interpretation makes the cosine function useful for describing circular motion.

Cosine on the Unit Circle

The unit circle, a circle with a radius of 1 centered at the origin, provides a powerful way to define cosine for any angle, even beyond the 0 to 90-degree range (or 0 to π/2 radians).

• At θ = 0: cos(0) = 1
• At θ = π/2: cos(π/2) = 0
• At θ = π: cos(π) = -1
• At θ = 3π/2: cos(3π/2) = 0
• At θ = 2π: cos(2π) = 1

The cosine function is periodic, meaning it repeats its values in regular intervals. Specifically, cos(θ) has a period of 2π, which means:

cos(θ + 2π) = cos(θ)

Graph of the Cosine Function

The graph of y = cos(θ) is a wave-like curve that oscillates between 1 and -1, repeating every 2π radians. Key characteristics of the cosine graph include:

• Amplitude: The height of the wave from the centerline (y = 0) to the peak. For y = cos(θ), the amplitude is 1.
• Period: The distance along the x-axis for one complete cycle of the wave. For cosine, the period is 2π.
• Phase Shift: The horizontal shift along the x-axis. The standard cosine function starts at (0,1), but phase shifts occur when the function is modified, for example cos(θ − φ), where φ is the phase shift.
• Vertical Shift: If a constant is added or subtracted, the entire graph moves up or down.

Properties of the Cosine Function

• Even Function: Cosine is an even function, meaning that cos(−θ) = cos(θ). This symmetry reflects across the y-axis.
• Range: The values of the cosine function are always between -1 and 1, i.e., −1 ≤ cos(θ) ≤ 1.
• Periodicity: As mentioned earlier, the cosine function repeats every 2π radians. This periodic nature is essential for modeling cyclic phenomena like sound waves or seasonal patterns.

Applications of the Cosine Function

The cosine function is applied in numerous fields, including:

• Physics: To describe harmonic motion, waveforms, and alternating current (AC) circuits.
• Engineering: For analyzing signal processing, vibrations, and structures involving oscillations.
• Geometry and Trigonometry: In the calculation of distances, angles, and for solving triangles.
• Astronomy: For modeling planetary orbits, which are often elliptical but can be approximated using circular functions.
• Music and Acoustics: Cosine functions are used to describe sound waves and harmonics.

Cosine in Terms of Euler's Formula

The cosine function can also be expressed using Euler's formula, which connects trigonometry and complex numbers:

e^iθ = cos(θ) + i sin(θ)

From this, we can extract:

cos(θ) = (e^iθ + e^−iθ)/2

This representation is particularly useful in more advanced mathematics and physics, especially in the study of complex waveforms and quantum mechanics.