뒤로Trigonometric Functions and Identities: Study Guide
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Trigonometric Functions and Identities
Fundamental Trigonometric Functions
Trigonometric functions are essential tools in mathematics, describing relationships between angles and sides in right triangles and periodic phenomena. The six primary trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant.
Sine (sen or sin): Measures the ratio of the opposite side to the hypotenuse in a right triangle.
Cosine (cos): Measures the ratio of the adjacent side to the hypotenuse.
Tangent (tg or tan): Ratio of sine to cosine.
Cotangent (cot): Reciprocal of tangent.
Secant (sec): Reciprocal of cosine.
Cosecant (cosec): Reciprocal of sine.
Formulas:
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. They are fundamental for simplifying expressions and solving equations.
Pythagorean Identity:
Double Angle Formulas:
Sum and Difference Formulas:
Reciprocal and Quotient Relationships
Each trigonometric function has a reciprocal, and some can be expressed as quotients of others.
Function | Reciprocal | Quotient Form |
|---|---|---|
Sine () | Cosecant () | --- |
Cosine () | Secant () | --- |
Tangent () | Cotangent () | |
Cotangent () | Tangent () |
Examples and Applications
Example 1: Find and .
Example 2: Use the double angle formula to find .
Summary Table of Key Trigonometric Identities
Identity | Formula |
|---|---|
Pythagorean | |
Double Angle (Sine) | |
Double Angle (Cosine) | |
Sum Formula (Cosine) | |
Difference Formula (Cosine) | |
Tangent (Sum) |
Additional info: Some formulas and terminology were inferred from context and standard trigonometric identities. The notation "sen" is used for sine, which is common in some languages (e.g., Portuguese, Spanish).
