Trigonometric Formulas and Relationships

This page presents a comprehensive overview of essential trigonometric formulas and relationships, serving as a valuable reference for students studying trygonometria. The content is organized into several key sections, each focusing on different aspects of trigonometric functions and their properties.

The first section introduces wzory redukcyjne, which are reduction formulas for trigonometric functions. These formulas are crucial for simplifying trigonometric expressions and solving complex problems.

Example: sin$90°+a$=cosa, cos$90°+a$=-sina

These examples demonstrate how angles greater than 90° can be reduced to expressions involving complementary angles.

The document then presents general trigonometric identities, known as wzory ogólne. These fundamental relationships form the basis of many trigonometric calculations and proofs.

Highlight: sin²a + cos²a = 1

This identity, known as the Pythagorean identity, is one of the most important in trigonometry, relating the squares of sine and cosine functions.

The guide also includes definitions of tangent and cotangent in terms of sine and cosine, which are essential for understanding the relationships between different trigonometric functions.

Definition: tga = sina / cosa, ctga = cosa / sina

These definitions help students understand how tangent and cotangent relate to the more fundamental sine and cosine functions.

The document provides a brief review section, reinforcing the reciprocal relationship between tangent and cotangent.

Vocabulary: KĄT ROZLEGŁY - This term refers to an obtuse angle, which is any angle greater than 90° but less than 180°.

The final section of the page deals with trigonometric ratios in the coordinate system, presenting formulas for sine, cosine, and tangent in terms of x and y coordinates. This connects trigonometric functions to analytical geometry, demonstrating their practical applications.

Example: sina = y / $√x² + √y²$, cosa = x / $√x² + √y²$

These formulas show how trigonometric ratios can be calculated from the coordinates of a point on the unit circle, bridging the gap between trigonometry and coordinate geometry.

The page concludes with additional wzory redukcyjne for angles near 180°, further expanding students' ability to work with angles in different quadrants of the coordinate system. Overall, this comprehensive guide serves as an excellent resource for students studying trigonometry, providing a solid foundation for more advanced mathematical concepts.