Introduction to Trigonometry in Geometry

Introduction to Trigonometry in Geometry

• Trigonometry is a branch of mathematics that deals with relationships between angles and sides of triangles.
• It's a fundamental tool in geometry, used to solve problems involving triangles, especially right-angled triangles.
• Key trigonometric ratios relate angles of a right-angled triangle to ratios of its sides.
• These ratios are sine, cosine, and tangent.

Trigonometric Ratios

• Sine (sin): Ratio of the side opposite the angle to the hypotenuse.
• Cosine (cos): Ratio of the side adjacent to the angle to the hypotenuse.
• Tangent (tan): Ratio of the side opposite the angle to the side adjacent to the angle.

Relationship between Ratios

• Ratios are related by identities; for example, sin²θ + cos²θ = 1.
• Trigonometric identities simplify complex expressions and equations.

Application of Trig Ratios in Solving Right Triangles

• Given one side length and one acute angle, use trig ratios to find other side or angle lengths.
• Solving for missing sides or angles in a right triangle is a key application.

Calculating Length of Sides

• Choose sine, cosine, or tangent based on known and needed sides.
• Establish a relationship between known and unknown sides using the appropriate ratio.
• Isolate the unknown side in the equation to calculate its value.

Calculating Angles

• Use inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹) if side lengths are known.
• Inverse functions find the angle corresponding to a given ratio.
• Example: sin⁻¹(opposite/hypotenuse) gives the angle.

Special Right Triangles

• 30-60-90 and 45-45-90 triangles have specific side ratios.
• Knowing these ratios simplifies calculations.

Trigonometric Functions

• Trigonometric functions are defined for all angles, not just acute angles in right triangles.
• Functions can be applied to angles outside 0° to 90°.
• Periodicity and graphs of trigonometric functions are crucial concepts.

Applications in Real-World Problems

• Many real-world problems use trigonometric functions.
• Examples include surveying (distances and heights), navigation (courses and distances), and engineering (structures and systems).
• Calculating heights of buildings or distances across water uses trigonometric principles.

Further Study Considerations

• Unit circle representations, including radian measure.
• Exploring trigonometric identities beyond basic ones.
• Investigating the behavior of trigonometric functions across different domains.