Topic: Mathematics > Trigonometry > Functions

Mathematics\Trigonometry\Functions

Trigonometric functions are fundamental mathematical functions that relate the angles of a triangle to the lengths of its sides. These functions are widely used in various branches of mathematics, physics, engineering, and many other fields. They are essential for understanding periodic phenomena, wave behavior, and oscillatory motion.

The primary trigonometric functions are:

1. Sine (sin): For a given angle \(\theta\) in a right-angled triangle, the sine function is defined as the ratio of the length of the opposite side to the hypotenuse.
   \[
   \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
   \]

2. Cosine (cos): The cosine function is the ratio of the length of the adjacent side to the hypotenuse.
   \[
   \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
   \]

3. Tangent (tan): The tangent function is the ratio of the sine function to the cosine function, or equivalently, the ratio of the opposite side to the adjacent side.
   \[
   \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\text{opposite}}{\text{adjacent}}
   \]

These functions have counterparts known as the reciprocal trigonometric functions:

1. Cosecant (csc): The reciprocal of the sine function.
   \[
   \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{\text{hypotenuse}}{\text{opposite}}
   \]

2. Secant (sec): The reciprocal of the cosine function.
   \[
   \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\text{hypotenuse}}{\text{adjacent}}
   \]

3. Cotangent (cot): The reciprocal of the tangent function.
   \[
   \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\text{adjacent}}{\text{opposite}}
   \]

Trigonometric functions can also be defined for any real number using the unit circle, a circle with a radius of one centered at the origin in the coordinate plane. Considering the unit circle allows extending trigonometric functions to all real numbers and provides a geometric interpretation that aids in understanding their properties and behaviors.

Unit Circle Definition:

• Sine of an angle \(\theta\) is the y-coordinate of the point on the unit circle corresponding to \(\theta\).
• Cosine of \(\theta\) is the x-coordinate.
• Tangent of \(\theta\) can be visualized as the length of the segment from the origin to the point where the terminal side of \(\theta\) intersects the vertical line \(x=1\).

\[
\begin{array}{cc}
\sin(\theta) = \text{y-coordinate} \\
\cos(\theta) = \text{x-coordinate} \\
\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \\
\end{array}
\]

The trigonometric functions are periodic, with sine and cosine having a period of \(2\pi\) and tangent having a period of \(\pi\). This means that the functions repeat their values in regular intervals, which is crucial when dealing with waves, oscillations, and circular motion.

Identities and Equations:

Trigonometric functions satisfy several important identities and equations, including:

1. Pythagorean Identity:
   \[
   \sin^2(\theta) + \cos^2(\theta) = 1
   \]

2. Angle Sum and Difference Identities: For any angles \(\alpha\) and \(\beta\),
   \[
   \sin(\alpha \pm \beta) = \sin(\alpha) \cos(\beta) \pm \cos(\alpha) \sin(\beta)
   \]
   \[
   \cos(\alpha \pm \beta) = \cos(\alpha) \cos(\beta) \mp \sin(\alpha) \sin(\beta)
   \]

3. Double Angle Formulas:
   \[
   \sin(2\theta) = 2\sin(\theta)\cos(\theta)
   \]
   \[
   \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)
   \]

These identities and properties of trigonometric functions are essential tools for solving equations and modeling periodic phenomena in both pure and applied mathematics. Understanding these functions opens the door to advanced concepts in calculus, complex analysis, and many applied sciences.