Trigonometric Basics: An Introduction to Angles and Functions

Branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides.

Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. The word "trigonometry" comes from the Greek words "trigonon" meaning triangle, and "metron" meaning measure. It has applications in various fields such as physics, engineering, computer science, and more.

Introduction to Trigonometry

Trigonometry is a mathematical method used for defining relations between elements of a triangle. In the world of mathematics, the Pythagorean theorem (which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides) is a classic example of trigonometry.

Angles

In trigonometry, we primarily deal with the angles and rotations. An angle is defined as a figure formed by two rays sharing a common endpoint, called the vertex. Angles are usually measured in degrees (°) or radians (rad).

Degree and Radian Measure

There are two units of measure for angles: degrees and radians. One full rotation (360°) is equivalent to 2π radians, and therefore, 180° is equivalent to π radians. To convert from degrees to radians, we multiply by π/180. Conversely, to convert from radians to degrees, we multiply by 180/π.

Trigonometric Functions of Acute Angles

Trigonometry introduces us to the trigonometric functions, which are used to relate the angles of a triangle to the lengths of its sides. The primary trigonometric functions are sine, cosine, and tangent, often abbreviated as sin, cos, and tan respectively.

Sine, Cosine, and Tangent

In a right triangle, the sine of an angle is the length of the opposite side divided by the length of the hypotenuse. The cosine of an angle is the length of the adjacent side divided by the length of the hypotenuse. The tangent of an angle is the length of the opposite side divided by the length of the adjacent side.

Reciprocal Trigonometric Functions

The reciprocal of sine is cosecant, of cosine is secant, and of tangent is cotangent. These are usually abbreviated as csc, sec, and cot respectively.

SOHCAHTOA

SOHCAHTOA is a mnemonic device that is used in mathematics to remember the definitions of the trigonometric functions of sine, cosine, and tangent. It stands for Sine equals Opposite over Hypotenuse, Cosine equals Adjacent over Hypotenuse, Tangent equals Opposite over Adjacent.

In conclusion, understanding the basics of trigonometry is crucial for further study in mathematics, especially in calculus. The concepts of angles, their measurement, and trigonometric functions form the foundation of trigonometry.