Trigonometric functions are mathematical functions that relate the ratios of the sides of a right triangle to the angles within the triangle. These functions are commonly denoted as sin, cos, tan, cot, sec, and csc. In this lesson, we will explore the definitions and properties of these six basic trigonometric functions.

Sine Function: The sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse of a right triangle. It is denoted by sin θ, where θ is the angle.

Cosine Function: The cosine of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse of a right triangle. It is denoted by cos θ, where θ is the angle.

Tangent Function: The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side of a right triangle. It is denoted by tan θ, where θ is the angle.

Cotangent Function: The cotangent of an angle is defined as the reciprocal of the tangent of the angle. It is denoted by cot θ, where θ is the angle.

Secant Function: The secant of an angle is defined as the reciprocal of the cosine of the angle. It is denoted by sec θ, where θ is the angle.

Cosecant Function: The cosecant of an angle is defined as the reciprocal of the sine of the angle. It is denoted by csc θ, where θ is the angle.

Trigonometric functions are used in a wide range of applications, including physics, engineering, and navigation. They are also used extensively in mathematics, especially in calculus and other branches of advanced mathematics.

In summary, the six basic trigonometric functions relate the ratios of the sides of a right triangle to the angles within the triangle. These functions are sine, cosine, tangent, cotangent, secant, and cosecant. They are used in a wide range of applications and are an important part of mathematics.