Trigonometric Functions

DEFINITION AND CONNECTION TO THE UNIT CIRCLE

Trigonometric functions are functions of an angle that calculate ratios between sides of a right-angled triangle or coordinate values on the unit circle. They are defined for real angles (in radians or degrees) and repeat periodically. The basic trigonometric functions are:

• sine (sin)
• cosine (cos)
• tangent (tan)
• cotangent (cot)

All four are defined for different sets of angle values, with tangent and cotangent having specific points where they are not defined (where division by zero occurs).

GEOMETRIC DEFINITION

In a right-angled triangle with an acute angle α:

• sin α = opposite side / hypotenuse
• cos α = adjacent side / hypotenuse
• tan α = opposite side / adjacent side
• cot α = adjacent side / opposite side

These definitions are valid for angles between 0° and 90° or 0 and π/2 radians. For broader values, trigonometric functions are defined using the unit circle.

UNIT CIRCLE

On the unit circle (radius = 1), for an angle α:

• A point on the circle has coordinates (cos α, sin α).
• tan α = sin α / cos α, if cos α ≠ 0.
• cot α = cos α / sin α, if sin α ≠ 0.

The functions are defined for all real angles, except at points where division by zero occurs.

BASIC PROPERTIES

• sin α and cos α are bounded between –1 and 1.
• tan α and cot α are unbounded.
• sin α is an odd function (sin(-α) = -sin(α)), cos α is an even function (cos(-α) = cos(α)).
• The functions are periodic:
• • sin and cos: period 2π
  • tan and cot: period π

VALUES FOR SPECIAL ANGLES

For angles 0, π/6 (30°), π/4 (45°), π/3 (60°), π/2 (90°), the values are known and frequently used:

• sin(π/6) = 1/2, cos(π/6) = √3/2
• sin(π/4) = √2/2, cos(π/4) = √2/2
• sin(π/3) = √3/2, cos(π/3) = 1/2

From these, tan and cot can be calculated where defined.

TRIGONOMETRIC IDENTITIES

• PYTHAGOREAN IDENTITY:
  sin² α + cos² α = 1
• TANGENT AND SINE/COSINE:
  tan α = sin α / cos α
• COTANGENT AND COSINE/SINE:
  cot α = cos α / sin α
• SYMMETRY PROPERTIES:
  sin(–α) = –sin α, cos(–α) = cos α

CONCLUSION

Trigonometric functions describe the basic ratios between the sides of a right-angled triangle or values on the unit circle. They are periodic, continuous (except where not defined), and fundamental for trigonometric analysis. Their behavior is precisely determined by symmetry, identities, and their connection to the coordinate system.