Page 2: Advanced Applications and Problem Solving

This page focuses on applying trigonometric concepts to solve complex problems involving Trójkąt prostokątny wzór na pole and diagonal calculations in rectangles.

Example: A problem involving a rectangle with sides of length 1 and 2, where the diagonal creates two triangles. The task involves calculating:

1. The lengths of the legs using Twierdzenie Pitagorasa
2. All trigonometric values (sine, cosine, tangent, cotangent) for both acute angles

Highlight: When using the Pythagorean theorem (a² + b² = c²), it's crucial to correctly identify the hypotenuse and legs.

Vocabulary: Przekątna (diagonal) - A line segment that connects two non-adjacent vertices of a polygon.

Example: For a right triangle with sides 3, 4, and 5:

• sin(α) = 3/5
• cos(α) = 4/5
• tan(α) = 3/4
• cot(α) = 4/3

Page 1: Fundamental Trigonometric Functions and Examples

This page introduces the core concepts of trigonometric functions in right triangles and provides detailed examples of their calculations.

Definition: The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.

Definition: The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.

Definition: The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.

Definition: The cotangent of an angle is the ratio of the length of the adjacent side to the length of the opposite side.

Highlight: In sine and cosine calculations, the result must always be a fraction, never a whole number. However, tangent and cotangent can result in whole numbers.

Example: For a 45° angle:

• sin(45°) = 1/√2
• cos(45°) = 1/√2
• tan(45°) = 1
• cot(45°) = 1

Example: For a 30° angle:

• sin(30°) = 1/2
• cos(30°) = √3/2
• tan(30°) = 1/√3
• cot(30°) = √3