Page 2: Trigonometric Functions and Right Triangle Trigonometry

This page delves into the fundamentals of trigonometric functions, particularly focusing on sine and cosine in the context of right triangles.

The page introduces the mnemonic device "CAH SOH TOA" to remember the definitions of cosine, sine, and tangent in right triangles. It then provides a detailed explanation of how to derive the values of sine and cosine for a 45-degree angle using the properties of isosceles right triangles.

Vocabulary: CAH SOH TOA is a mnemonic where:

• CAH: Cosine = Adjacent / Hypotenuse
• SOH: Sine = Opposite / Hypotenuse
• TOA: Tangent = Opposite / Adjacent

Example: In a 45-45-90 triangle, the ratio of the legs to the hypotenuse is derived using the Pythagorean theorem, resulting in the well-known √2/2 value for both sine and cosine of 45°.

The page emphasizes the importance of understanding these basic trigonometric ratios as they form the foundation for more complex trigonometric concepts.

Page 3: Notable Trigonometric Values and Function Properties

This page presents a comprehensive table of valeurs remarquables trigonométrie (notable trigonometric values) and discusses the properties of sine and cosine functions.

The table includes the cosine and sine values for angles 0°, 30°, 45°, 60°, 90°, and 180°, both in fraction and decimal forms. These values are crucial for solving trigonometric problems without a calculator.

Highlight: Memorizing these notable values is essential for efficient problem-solving in trigonometry.

The page then provides graphical representations of the sine and cosine functions, highlighting their periodic nature and symmetry properties.

Definition: Symmetry in trigonometric functions:

• The sine function is odd (symmetric about the origin)
• The cosine function is even $symmetric about the y-axis$

Key properties of sine and cosine functions are listed, including their behavior with negative angles and angles greater than π.

Page 4: Advanced Trigonometric Properties and Identities

This page expands on the properties of trigonometric functions, introducing more complex relationships and identities.

The page presents several important trigonometric identities, including those involving complementary angles and the fundamental trigonometric identity sin²x + cos²x = 1.

Highlight: The identity sin²x + cos²x = 1 is fundamental in trigonometry and is used in many proofs and problem-solving scenarios.

The page also introduces the concept of periodicity in trigonometric functions, showing how sine and cosine values repeat every 2π radians.

Definition: Periodicity in trigonometric functions refers to the repetition of values over regular intervals. For sine and cosine, this period is 2π radians or 360°.

These properties and identities are essential for solving more complex trigonometric equations and for understanding the behavior of trigonometric functions in various contexts.

Page 5: Deriving Trigonometric Values for 30° and 60°

This page provides a detailed derivation of the sine and cosine values for 30° and 60° angles using geometric reasoning and the properties of equilateral triangles.

The derivation starts with an equilateral triangle and uses its properties to construct a 30-60-90 triangle. Through step-by-step geometric and algebraic reasoning, the page shows how to arrive at the standard values for sine and cosine of these angles.

Example: For a 60° angle in a 30-60-90 triangle:

• cos$60°$ = 1/2
• sin$60°$ = √3/2

Highlight: Understanding these derivations helps in memorizing the valeurs remarquables trigonométrie and provides insight into the geometric meaning behind these values.

The page emphasizes the importance of the Pythagorean theorem and the properties of special right triangles in deriving these trigonometric values.

Page 6: Additional Trigonometric Calculations

The final page appears to contain additional calculations and examples related to trigonometric functions, particularly focusing on the sine and cosine of 30° angles.

While the content on this page is less structured than the previous pages, it seems to reinforce the concepts introduced earlier, providing more practice with trigonometric calculations.

Example: The page shows calculations for sin$30°$ = 1/2 and cos$30°$ = √3/2, which are important valeurs remarquables trigonométrie.

This page serves as a practical application of the concepts learned throughout the document, allowing students to see how the theoretical knowledge translates into actual problem-solving.

Page 1: Angle Conversions and Trigonometric Circle

This page introduces the conversion between degrees and radians, as well as the concept of the trigonometric circle.

The page begins by presenting formulas for converting angles from degrees to radians and vice versa. It emphasizes the relationship between degrees and pi (π) in these conversions. The trigonometric circle is then introduced, showing how angles are represented on it.

Definition: The radian is the standard unit of angular measure in many areas of mathematics. One radian is the angle subtended at the center of a circle by an arc equal in length to the radius of the circle.

Example: To convert 150° to radians, the formula y = (x × π) / 180 is used, where x is the angle in degrees and y is the angle in radians.

Highlight: The relationship between degrees and radians is crucial in trigonometry, with 180° being equivalent to π radians.

The page also includes a section on finding points on the trigonometric circle corresponding to specific angles, both in degrees and radians.