Proving the Trigonometric Identity: tanx sinx / cosx

Introduction

The relationship between the trigonometric functions sine, cosine, and tangent is fundamental in trigonometry. This article delves into the proof of the identity tanx sinx / cosx. Understanding this identity is crucial in various applications, ranging from calculus to physics.

Right Triangle Definitions

Let's consider a right triangle ABC with the right angle at C. Suppose angle A is a non-right angle.

Hypotenuse (H): The longest side of a right triangle, opposite the right angle. Opposite (O): The side opposite angle A. Adjacent (A): The side next to angle A, which is not the hypotenuse.

Trigonometric Definitions

Sine (sinx): defined as the length of the opposite side divided by the length of the hypotenuse. Mathematically, sin x O / H. Cosine (cosx): defined as the length of the adjacent side divided by the length of the hypotenuse. Mathematically, cos x A / H. Tangent (tanx): defined as the length of the opposite side divided by the length of the adjacent side. Mathematically, tan x O / A.

Proof Using Right Triangle Definitions

Consider a right triangle ABC with angle A and sides opposite A as O, adjacent to A as A, and hypotenuse as H.

By the definition of the functions involved:

Tangent (tanx)

Tan x O / A

Sine (sinx) and Cosine (cosx)

Using the definitions of sine and cosine, we have:

Sin x O / H Cos x A / H

Therefore:

Tan x O / A (O / H) / (A / H) Sin x / Cos x

Proof Using Unit Circle Representations

The proof can be extended to use the unit circle approach, which is a powerful tool in trigonometry.

Unit Circle

In a unit circle, the hypotenuse H is always 1. Thus:

Sin x O / H O (since H 1) Cos x A / H A (since H 1) Tan x O / A

Rewriting it using the definitions of sine and cosine:

Tan x O / A Sin x / Cos x

BASIC Identity and Mnemonics

The relationship tanx sinx / cosx is a basic identity in trigonometry and does not require proof as it is a direct consequence of the definitions of sine and cosine.

SOHCAHTOA

A mnemonic to help remember the basic definitions:

  S
  in   O
  pposite  H
  ypotenuse
  C
  os   A
  djacent  H
  ypotenuse
  T
  an   O
  pposite  A
  djacent
  S
  ome   O
  wners'  H
  orses  C
  an   A
  lways   H
  ere  T
  he  O
  wners  A
  pproaching

Conclusion

In summary, the identity tanx sinx / cosx is a fundamental property of trigonometric functions and can be derived from the definitions of sine and cosine. This identity is widely used in various scientific and technical fields, making it essential to understand its derivation and application.