Page 2: Advanced Trigonometric Functions

This page delves into transformations of trigonometric functions and essential identities.

Vocabulary: Cast diagrams are visual tools used to determine the signs of trigonometric ratios in different quadrants.

Definition: The fundamental trigonometric identity sin²θ + cos²θ = 1 forms the basis for many trigonometric proofs.

Example: When solving 2cosx = -1 in the range -180° ≤ x ≤ 540°, solutions are found at x = 120°, 240°

Page 3: Complex Trigonometric Applications

This section covers advanced topics including quadratic trigonometry and compound angles.

Highlight: Quadratic trigonometry factorization problems can be solved by treating trigonometric equations as standard quadratic equations.

Definition: Compound angle formulae:

• sin$A+B$ = sinA cosB + cosA sinB
• cos$A+B$ = cosA cosB - sinA sinB

Example: The equation tan²x - 2tanx = 0 can be factored as tanx$tanx - 2$ = 0

Page 4: Double Angle Formulae and R-Addition

This page explores double angle formulae and R-addition techniques.

Definition: Double angle formulae include:

• sin(2A) = 2sinA cosA
• cos(2A) = cos²A - sin²A

Example: For R-addition formula, 5cosx - 12sinx can be written as 13cos$x + 67.38°$

Highlight: The R-addition formula transforms expressions of the form asinx + bcosx into Rsin$x + α$ or Rcos$x + α$

Page 5: Advanced Trigonometric Proofs

The final page focuses on complex trigonometric proofs and properties.

Example: Proving relationships involving tangent and cotangent functions requires careful manipulation of basic identities.

Highlight: Advanced proofs often involve converting expressions to simpler forms using fundamental identities.

Definition: The relationship between tangent and cotangent functions is essential for solving complex trigonometric equations.

Page 1: Fundamental Trigonometric Concepts

This page introduces the core concepts of trigonometry, focusing on basic ratios and rules for triangle calculations.

Definition: Trigonometric ratios are defined as relationships between sides of right-angled triangles:

• sine = opposite/hypotenuse
• cosine = adjacent/hypotenuse
• tangent = opposite/adjacent

Highlight: The cosine rule $a² = b² + c² - 2bc cosA$ is essential for solving non-right angled triangles with cosine rule.

Example: Trigonometric graphs show the periodic nature of functions, with sin curves ranging between -1 and 1.