What is Trigonometry?

Explanation

Trigonometry, one of the fundamental branches of mathematics, focuses on the study of angles and their relationships with the sides of triangles. This article explains the key concepts of trigonometry that are essential for understanding this topic.

TRIGONOMETRY: UNDERSTANDING ANGLES AND TRIANGLES

ANGLES: Angles are the foundation of trigonometry. They are measured in degrees or radians and describe the amount of opening between two intersecting lines.
TYPES OF TRIANGLES: Trigonometry primarily focuses on right-angled triangles, where one angle measures 90 degrees, and also deals with other types like scalene or isosceles triangles (the original mentioned equilateral, but the theorems apply more broadly).

TRIGONOMETRIC FUNCTIONS

SINE (sin x): The sine of an angle in a right-angled triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.
COSINE (cos x): The cosine of an angle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
TANGENT (tan x): The tangent of an angle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
COTANGENT (cot x): The cotangent of an angle is the ratio of the length of the side adjacent to the angle to the length of the side opposite that angle.

USE OF THE UNIT CIRCLE

The unit circle in trigonometry is used to visualize trigonometric functions such as sine, cosine, and tangent, where the radius of the circle is equal to 1. As the angle changes, a point on the circumference of the unit circle represents the terminal values of cosine and sine for that angle on the x- and y-axes, respectively. This model facilitates easier understanding and calculation of trigonometric functions and their properties, such as periodicity and symmetry.

TRIGONOMETRY AND SOLVING TRIANGLES

PYTHAGOREAN THEOREM: In a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides (legs).
SINE RULE (LAW OF SINES): The ratio of the length of a side to the sine of its opposite angle is the same for all sides and angles in any triangle.
COSINE RULE (LAW OF COSINES): This theorem allows for the calculation of sides and angles in any triangle.

CONCLUSION

The basics of trigonometry provide a fundamental understanding of the relationships between angles and sides in triangles. This knowledge is crucial for students, engineers, scientists, and anyone dealing with technical and geometric problems. Understanding these basics is the first step towards a deeper exploration of the mathematical world.