Introduction to Trigonometry

Study Notes

Introduction to Trigonometry

Trigonometry is a branch of mathematics that studies relationships between angles and sides of triangles.
It's crucial in various fields, including navigation, surveying, engineering, and physics.
Fundamental concepts include angles, sides, and ratios of right-angled triangles.

Key Trigonometric Ratios

Sine (sin): The ratio of the side opposite to an acute angle to the hypotenuse.
Cosine (cos): The ratio of the side adjacent to an acute angle to the hypotenuse.
Tangent (tan): The ratio of the side opposite to an acute angle to the side adjacent to it.
These ratios are often expressed as abbreviations, for example sin θ = opposite/hypotenuse

Relationship between Trigonometric Ratios

The Pythagorean Theorem (a² + b² = c²) relates the three sides of a right-angled triangle.
Trigonometric ratios are interconnected and can be used to find missing sides or angles in right-angled triangles.

Determining trigonometric values for angles

Special angles (30°, 45°, 60°): Exact values for sine, cosine, and tangent are known and derived from the ratios of sides in specific right triangles.
Using calculators: Calculators can be used to find trigonometric values for any angle.

Applications of Trigonometry

Solving right-angled triangles: Trigonometry tools can find unknown sides or interior angles of a right triangle if some parameters are known.
Calculating heights and distances: Trigonometry makes it possible to determine inaccessible heights and distances.
Navigation: Trigonometry plays a fundamental part in determining location and direction.
Surveying: Surveyors use trigonometric calculations to identify and measure land areas.
Engineering: Engineers utilize trigonometric concepts in structural design and analyses involving angles and distances.
Physics: Trigonometry is deeply interwoven into physics, particularly in scenarios involving vectors and projectile motion.

Trigonometric Functions

sin θ, cos θ, tan θ: These basic functions relate the angles of a triangle to the ratios of its sides.
Reciprocal trigonometric functions (cosecant, secant, cotangent): These functions are reciprocals of the basic trigonometric functions and have importance in various applications.

Trigonometric Identities

Fundamental identities: Establish relationships between trigonometric functions—e.g., sin²θ + cos²θ = 1. These identities are essential for simplifying and solving trigonometric equations.
Derived identities: Built upon fundamental identities and are crucial for performing advanced trigonometric manipulations. They aid in simplifying expressions and in solving trigonometric equations.

Graphing Trigonometric Functions

Periodic nature: These functions graph as periodic waves, repeating at regular intervals. Understanding the patterns is critical to analyze their behavior.
Domain and range of functions: The applicable input values (domain) and output values (range) describe the behavior of trigonometric functions.

Inverse Trigonometric Functions

arcsin, arccos, arctan: These functions find the angle associated with a given trigonometric ratio.
Principal values: The corresponding angle in a defined range for each inverse.
These are used to find angles when the trigonometric ratio is known.

Description

Explore the fundamental concepts of trigonometry, including key trigonometric ratios like sine, cosine, and tangent. This quiz delves into the relationships between angles and sides of triangles, highlighting the importance of these ratios in various fields. Perfect for students looking to solidify their understanding of right-angled triangles.