unit circle

The unit circle is a circle of radius 1 centered at the origin, commonly used to define the trigonometric functions.

Sine and cosine are fundamental trigonometric functions that associate each angle, defined from the unit circle, with a value between -1 and +1.

In a right triangle, the tangent of an angle is defined as the ratio between the opposite and adjacent sides, while the cotangent is the reciprocal of this ratio. Both can also be expressed in terms of sine and cosine functions as:

\[
\begin{array}{cc}
\tan x = \dfrac{\sin x}{\cos x} & \quad \cot x = \dfrac{\cos x}{\sin x}
\end{array}
\]

The secant and cosecant of an angle \( \theta \) are defined, respectively, as the reciprocals of \( \cos\theta \) and \( \sin\theta \).

\[ \sec(\theta) = \frac{1}{\cos(\theta)} \quad \csc(\theta) = \frac{1}{\sin(\theta)}\]

The Pythagorean identity is an equation connecting trigonometry and geometry. The identity is expressed as:

\[
\sin^2\theta + \cos^2\theta = 1
\]

The method of reference angles refers to a set of trigonometric identities that allow one to express trigonometric functions of non-acute angles, represented on the unit circle, in terms of the corresponding acute angle in the first quadrant of the Cartesian coordinate system.

Trigonometric inequalities are inequalities involving sine, cosine, tangent, or related functions, solved using periodicity, reference angles, and the unit circle.

The cosine function \( f(x) = \cos(x) \) assigns to each angle \( x \), expressed in radians, its corresponding cosine value.