Cosine

Cosine

The cosine is a fundamental trigonometric function that is widely used in various fields of mathematics, physics, and engineering. It is one of the primary functions in trigonometry, alongside the sine and tangent functions. The cosine of an angle in a right-angled triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.

Definition

In a right-angled triangle, the cosine of an angle \( \theta \) is given by: \[ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]

Unit Circle Definition

The cosine function can also be defined using the unit circle. The unit circle is a circle with a radius of one, centered at the origin of a coordinate system. For an angle \( \theta \) measured from the positive x-axis, the coordinates of the point on the unit circle are \( (\cos(\theta), \sin(\theta)) \).

Properties

• Periodicity: The cosine function is periodic with a period of \( 2\pi \). This means that \( \cos(\theta + 2\pi) = \cos(\theta) \) for any angle \( \theta \).
• Even Function: The cosine function is an even function, which means that \( \cos(-\theta) = \cos(\theta) \).
• Range: The range of the cosine function is \([-1, 1]\).

Graph

The graph of the cosine function is a wave that oscillates between -1 and 1. It starts at 1 when \( \theta = 0 \), decreases to -1 at \( \theta = \pi \), and returns to 1 at \( \theta = 2\pi \).

Applications

The cosine function is used in various applications, including:

• Fourier series: In signal processing, the cosine function is used to represent periodic signals.
• Harmonic motion: In physics, the cosine function describes the motion of oscillating systems such as springs and pendulums.
• Vector mathematics: In vector mathematics, the cosine of the angle between two vectors is used to determine the dot product.

Related Functions

• Sine
• Tangent
• Secant
• Cosecant
• Cotangent

See Also

• Trigonometry
• Unit circle
• Pythagorean identity
• Euler's formula
• Inverse trigonometric functions