Introduction to Trigonometry

Defining Trigonometry

• Trigonometry is a branch of mathematics studying relationships between angles and sides of triangles.
• Widely used in engineering, navigation, astronomy, and computer graphics.
• Based on ratios of sides in right-angled triangles.

Basic Trigonometric Ratios

• Fundamental trigonometric ratios are sine (sin), cosine (cos), and tangent (tan).
• Defined for acute angles in a right triangle.
• sin(θ) = opposite/hypotenuse
• cos(θ) = adjacent/hypotenuse
• tan(θ) = opposite/adjacent

Understanding the Relationship

• Ratios depend on the angle's measure (θ).
• Changing the angle changes the trigonometric function's value.
• Memorizing the ratios helps determine unknown sides or angles in right triangles.

Reciprocal Trigonometric Ratios

• Reciprocal ratios are cosecant (csc), secant (sec), and cotangent (cot).
• They are reciprocals of the basic ratios.
• csc(θ) = 1 / sin(θ)
• sec(θ) = 1 / cos(θ)
• cot(θ) = 1 / tan(θ)

Special Triangles

• 30-60-90 and 45-45-90 triangles have specific side ratios.
• These special triangles allow quick calculation of trigonometric ratios for standard angles.
• Knowing these ratios simplifies problem-solving.

Angles of Elevation and Depression

• Angle of elevation: angle formed by looking up at an object from an observer.
• Angle of depression: angle formed by looking down at an object from an observer.
• Used in applications involving heights and distances (e.g., surveying, architecture, navigation).

Applications in Real-world scenarios

• Trigonometry is crucial for calculating heights of buildings, distances across rivers, and object angles.
• Used extensively in construction, surveying, and technical fields.
• Problem-solving requires identifying given information, unknowns, and applying appropriate trigonometric ratios.

Solving Right Triangles

• Appropriate trigonometric ratios are used to find unknown sides or angles.
• Clear understanding of the triangle's parts and given data is essential for correct calculations.
• Steps often involve drawing a diagram, labeling known values, choosing formulas, and then calculating unknowns.

Pythagorean Theorem

• Pythagorean theorem is frequently used in trigonometry problem-solving.
• Connects the sides of a right triangle, creating a core relationship.
• Can calculate unknown side lengths.

Trigonometric Identities

• Fundamental identities are crucial for simplifying complex expressions.
• Show relationships between different trigonometric functions/ratios.
• Mastering identities enables solving complex equations and identities.