Trigonometric Functions and Identities

Fundamental Trigonometric Functions

Trigonometric functions are essential in analyzing relationships between angles and sides in right triangles and in describing periodic phenomena. The six primary trigonometric functions are defined as follows:

• Sine (senx): Measures the ratio of the opposite side to the hypotenuse in a right triangle.

• Cosine (cosx): Measures the ratio of the adjacent side to the hypotenuse.

• Tangent (tgx): Measures the ratio of the opposite side to the adjacent side.

• Cotangent (cotx): Reciprocal of tangent.

• Secant (secx): Reciprocal of cosine.

• Cosecant (cossecx): Reciprocal of sine.

Formulas:

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables for which the functions are defined. They are fundamental tools for simplifying expressions and solving equations.

• Pythagorean Identity:

• Double Angle Formulas:

• Sum and Difference Formulas:

  • (Additional info: Standard formula for tangent difference)

Reciprocal and Quotient Relationships

Each trigonometric function has reciprocal and quotient relationships, which are useful for rewriting expressions and solving equations.

Trigonometric Function Table

The following table summarizes the relationships between the six trigonometric functions:

Examples and Applications

• Example 1: Find if and .

  • Using the double angle formula:

• Example 2: Simplify given , , , .

Additional info: Standard tangent sum and difference formulas were added for completeness.