Trigonometric Functions

Trigonometric functions are fundamental in the field of trigonometry, and they describe relationships between the angles and sides of triangles. They are also defined for complex numbers and are used in various areas of mathematics and science.

Basic Trigonometric Functions

  1. Sine (sin): In a right triangle, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.

  2. Cosine (cos): In a right triangle, the cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.

  3. Tangent (tan): In a right triangle, the tangent of an angle is the ratio of the sine of the angle to the cosine of the angle, which is equivalent to the ratio of the opposite side to the adjacent side.

Reciprocal Trigonometric Functions

  1. Cosecant (csc): The cosecant is the reciprocal of the sine. So, csc(θ) = 1/sin(θ).

  2. Secant (sec): The secant is the reciprocal of the cosine. So, sec(θ) = 1/cos(θ).

  3. Cotangent (cot): The cotangent is the reciprocal of the tangent. So, cot(θ) = 1/tan(θ).

Unit Circle Definition

Trigonometric functions can also be defined using a unit circle (a circle of radius 1 centered at the origin of a coordinate plane). For an angle θ:

  • sin(θ) is the y-coordinate of the point on the unit circle that is θ radians counterclockwise from (1, 0).
  • cos(θ) is the x-coordinate of that point.

Applications

Trigonometric functions are used in various fields, including physics, engineering, computer science, statistics, and more. They can model periodic phenomena (like sound or light waves), solve triangles in trigonometry, and solve problems in calculus, among other things.