Trigonometric Functions and Fundamental Identities

Basic Trigonometric Functions

Trigonometric functions relate the angles of a triangle to the lengths of its sides. The six primary trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent.

• Sine (sen or sin):

• Cosine (cos):

• Tangent (tg or tan):

• Cosecant (cosec or csc):

• Secant (sec):

• Cotangent (cot):

Fundamental Trigonometric Identity

The most important identity in trigonometry is the Pythagorean identity:

This identity is used to derive other trigonometric relationships.

Trigonometric Addition and Subtraction Formulas

Sum and Difference Formulas

These formulas allow the calculation of the sine, cosine, or tangent of the sum or difference of two angles.

• Cosine of Sum:

• Cosine of Difference:

• Tangent of Sum:

• Tangent of Difference:

Double Angle Formulas

Double angle formulas are used to express trigonometric functions of in terms of .

• Sine Double Angle:

• Cosine Double Angle:

• Tangent Double Angle:

Reciprocal and Quotient Identities

These identities relate the basic trigonometric functions to their reciprocals and quotients.

• Secant:

• Cosecant:

• Cotangent:

Example: Using the Double Angle Formula

Find if and is in the first quadrant.

• First, find using the Pythagorean identity:

• Then,

Summary Table: Trigonometric Functions and Their Reciprocals

Additional info: Some formulas and terms were inferred and clarified based on standard trigonometric identities and the context of the provided notes.