Why Are There Only Three Laws for Sine, Cosine, and Tangent Functions in Trigonometry?

When delving into the realm of trigonometry, one might wonder why the sine, cosine, and tangent functions are the main focus, given that essential relationships exist among other functions such as secant, cosecant, and cotangent. This article aims to explore the reasons behind the prevalence of sine, cosine, and tangent laws and provide a comprehensive understanding of these trigonometric functions.

Understanding the Triangle and Trigonometric Functions

Consider a right triangle with three sides: a, b, and c. In Greek trigonometry, a and b are the legs or catheti (plural of cathetus) adjacent to the right angle, while c is the hypotenuse (the longest side) opposite to the right angle. Let α be the angle opposite to side a, and β be the angle opposite to side . With this setup, several trigonometric relations emerge.

The Sine, Cosine, and Tangent Functions

The sine function (sin(α)) is defined as the ratio of the side opposite to the angle (side a) to the hypotenuse (c):

sin(α) a/c

The cosine function (cos(α)) is defined as the ratio of the adjacent side (side b) to the hypotenuse (c):

cos(α) b/c

The tangent function (tan(α)) is defined as the ratio of the opposite side (side a) to the adjacent side (side b):

tan(α) a/b

Additional Trigonometric Functions

While sine, cosine, and tangent are the primary trigonometric functions, other trigonometric ratios such as secant, cosecant, and cotangent are also useful. These functions are defined as follows:

Secant (sec(α)) is the reciprocal of cosine: sec(α) 1/cos(α) Cosecant (csc(α)) is the reciprocal of sine: csc(α) 1/sin(α) Cotangent (cot(α)) is the reciprocal of tangent: cot(α) 1/tan(α)

Practical Implications in Trigonometric Calculations

Modern calculators and computing devices typically provide built-in functions for sine, cosine, and tangent. However, this does not mean that secant, cosecant, and cotangent are unimportant. In fact, they are often used in various applications, such as in navigation, engineering, and physics.

Interrelations Among Trigonometric Functions

It is indeed true that these additional trigonometric functions can be expressed in terms of sine and cosine. For example: cotangent: cot(α) 1/tan(α) cos(α)/sin(α) secant: sec(α) 1/cos(α) cosecant: csc(α) 1/sin(α) c/b

These interrelations allow for versatility in problem solving and theoretical analysis. Understanding these functions and their relationships is crucial in advanced trigonometric studies and practical applications.