Page 2: Solving a Right Triangle Step-by-Step

This page demonstrates how to rozwiąż trójkąt prostokątny (solve a right triangle) with a practical example. The problem involves a right triangle where one angle is 30° and the adjacent side (przyprostokątna) has a length of 8.

The solution is presented in a step-by-step manner:

Step 1: Draw a helpful diagram and identify the given information.

Step 2: Recognize this as a 30°-60°-90° special right triangle or use trigonometry.

Step 3: Calculate the third angle of the triangle (180° - 30° - 90° = 60°).

Step 4: Determine the remaining sides using either the 30°-60°-90° triangle properties or trigonometric ratios.

Example: Using the 30°-60°-90° triangle properties, if the shortest side (opposite to 30°) is 4, then the hypotenuse is 8, and the remaining side is 4√3.

Highlight: The solution can be approached using either special triangle properties or trigonometric functions, demonstrating flexibility in problem-solving methods.

The page concludes with the complete solution: Angles: 30°, 60°, 90°; Sides: 8, 4, 4√3.

This practical example illustrates how to rozwiązywanie trójkątów prostokątnych zadania (solve right triangle problems) effectively by applying geometric and trigonometric principles.

Page 1: Fundamentals of Solving Right Triangles

This page introduces the concept of rozwiązywanie trójkątów prostokątnych (solving right triangles) and reviews key mathematical principles. To solve a right triangle means to determine all side lengths and angle measures.

Definition: Solving a right triangle involves finding the lengths of all sides and the measures of all angles.

The page reviews three fundamental concepts:

1. The Pythagorean Theorem: a²+b²=c², where c is the hypotenuse and a and b are the other two sides.

2. The sum of angles in a triangle: α + β + γ = 180°

3. Trigonometric functions: sine, cosine, tangent, and cotangent ratios for angles in a right triangle.

Highlight: Special right triangles (30°-60°-90° and 45°-45°-90°) have fixed side length ratios, making calculations easier.

The page includes diagrams illustrating these concepts and provides formulas for trigonometric functions in these special triangles.

Example: In a 30°-60°-90° triangle, if the shortest side is a, the hypotenuse is 2a, and the remaining side is a√3.

This foundational knowledge is crucial for jak rozwiązać trójkąt (how to solve a triangle) efficiently.