Trigonometric Identities and Trigonometry Formulas: A Comprehensive Guide


Humans and mathematics are today regarded as two sides of the same coin. That is, without one another, they are incomplete. Mathematics has made human life easier and more efficient. It is said to be a topic where scholars can discover a range of new things. People have begun to feel that if they do not understand basic mathematics, they will have a tough time surviving on this earth. Mathematics is one of those courses that the majority of pupils despise. However, this is not the case; if a student works hard by practicing regularly and dedicating himself to the subject, he will be able to easily target this subject provided he understands all of the fundamental ideas. There are numerous options.

Trigonometric Formulas
Trigonometric Formulas

A trigonometric identity is a mathematical equation that states that two specific trigonometric ratios are equal. Trigonometric ratios are three quantities in a right triangle that relate to each other through the law of sines and cosines. This means that the ratio of the lengths of any two sides of a triangle, or the ratio of any side’s length to another side’s length, can be determined by knowing only the angles opposite these sides. The most common trigonometric ratios are sine (sin), cosine (cos), and tangent (tan). A relationship between a specific trigonometric ratio and its inverse also exists; like all trigonometric ratios, this is also a ratio of the lengths of two sides of a triangle.

The common trigonometric identities hold true for a right triangle. A special class of identities is Pythagorean identities which describe relations between the three functions sine, cosine, and tangent. Trigonometry formulas deal with relations between the six trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent.

An important relationship is a Pythagorean theorem which asserts that in a triangle with two acute angles where one side has length “c” and the other has length “b” and an obtuse angle has a length of “a”, sin(A) = (c/a) and tan(A)= (c/b). This can be applied to find the sine, cosine, or tangent of the sum or difference of two angles. 

Some common trigonometry formulas are:

  • sin θ = Opposite Side/Hypotenuse.
  • cos θ = Adjacent Side/Hypotenuse.
  • tan θ = Opposite Side/Adjacent Side.
  • sec θ = Hypotenuse/Adjacent Side.
  • cosec θ = Hypotenuse/Opposite Side.
  • cot θ = Adjacent Side/Opposite Side.

The trigonometric functions used may be defined by more than one formula; for example, cotangent is defined as 1 divided by tangent. In such cases, it is usually best to use the simplest possible form that still gives a correct result. For instance, if a problem asks for the inverse sine then it would not be correct to use the version of inverse sine. But it would be correct to substitute in the definition of inverse cosine instead.

Here are some of the formula which one studies in trigonometry:

  1. sin^2+cos^2=1
  2. sec^2-tan^2=1

The trigonometric identities are useful for simplifying some more complex formulas.


  • To find the trigonometric ratios of angles
  • To simplify algebraic equations involving trigonometric functions.
  • To prove trigonometric identities.
  • In complex formulas.

These are some of the main uses of Trigonometry.

This was some basic information on trigonometric identities and formulas and if one wants more information all they have to do is explore the cuemath website.

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Robert Keith

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