Trigonometric Identities and Functions

Basic Trigonometric Functions

Trigonometric functions relate the angles of a triangle to the lengths of its sides. The six primary trigonometric functions are:

• Sine (sen or sin):

• Cosine (cos):

• Tangent (tg or tan):

• Cotangent (cot):

• Secant (sec):

• Cosecant (cosec or csc):

Fundamental Trigonometric Identity

The most important identity in trigonometry is the Pythagorean identity, which relates sine and cosine:

• Pythagorean Identity:

Sum and Difference Formulas

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

• Cosine of Sum:

• Cosine of Difference:

• Tangent of Sum:

• Tangent of Difference: (Additional info: Not explicitly in the notes, but included for completeness.)

Example: To find , use and apply the sum formula.

Double Angle Formulas

Double angle formulas express trigonometric functions of in terms of .

• Sine Double Angle:

• Cosine Double Angle:

• Tangent Double Angle: (Additional info: Not explicitly in the notes, but included for completeness.)

Example: If and is in the first quadrant, find using the double angle formula.

Reciprocal and Quotient Identities

These identities relate the trigonometric functions to each other through reciprocals and quotients.

• Reciprocal Identities:

• Quotient Identities:

Table: Trigonometric Functions and Their Relationships

Additional info:

• Some formulas and notations were inferred and clarified for academic completeness.

• "Sen" and "cos" are the Portuguese/Spanish notations for "sin" and "cos".

• Some formulas (e.g., tangent of sum/difference, double angle for tangent) were added for completeness, as they are standard in trigonometry.