Trigonometric Formulae-Important Formulae

There are main six types of trigonometric ratios or functions that are used to make the trigonometric identities. Sine, cosine, tangents are the basic trigonometric identities. All these trigonometric ratios are defined on a right angles triangle. A right-angle triangle consists of a perpendicular, a base, and a hypotenuse. The angle between the perpendicular and the base is a right angle that is 90 degrees. Based on the other two angles, these trigonometric ratios have been defined. All the trigonometric identities are made up of using these ratios and are very useful for solving some hard problems.

Identities of the trigonometric formulae

There are so many identities based on trigonometric ratios. At first, the trigonometric ratios are defined using the three sides of a right-angled triangle. Consider a right angle triangle with three sides AB, BC, and CA. The right angle is placed between the side AB and BC. That means angle B is the right angle of triangle ABC. So, the side AB is called perpendicular, BC as a base, and AC is the hypotenuse. Trigonometric ratios are defined as “sin = Perpendicular/Hypotensue, cos = Base/Hypotensue, tan= Perpendicular/Base,  cosec= Hypotensue/Perpendicular, sec= Hypotensue/Base, and cot= Base/Perpendicular”. Based on these trigonometric ratios, there are so many trigonometric identities that are very useful and important for solving many problems. Using the Pythagoras formula for a right-angle triangle, there are main three types of trigonometric identities. Pythagoras formula stated that “Perpendicular²+Base²= Hypotensue²”

Using the above formula, there are three basic trigonometric identities. Such as” sin²θ+cos²θ =1, sec²θ+tan²θ=1″, and “cosec²θ-cot²θ=1″. All these basic identities can be proved using the Pythagoras formula of right-angle triangle. Besides these formulae, there are so many formulae that are using complementary angle concepts. If the sum of two angles is 90 degrees, then the two angles are known as complements to each other. In these trigonometry ratios, some ratios are a complement to some other ratios, and vice-versa.  Such as, “sinθ =cos (90-θ) , tanθ = (90-θ) “, and θ = sec (90-θ)” . Based on these basic identities, so many basic problems can be solved, and further identities were also made. There are some reciprocal identities too. Such as “sinθ =1/ cosθ, cosθ=1/sinθ , and tanθ =1/cotθ ”.

Expression of trigonometric formulae

Depending on the basic trigonometric identities, there are so many advanced identities that are very helpful to solve major mathematical problems. In order to find the formulae for compound angle, which is a sum of two other angles, trigonometric ratios are very useful and important. There are so identities for compound angle too, such as “sin(A+B) = sinAcosB + cosAsinB”, and many more. These formulae are used to solve problems based on compound angles of a triangle. All these identities are based on the basic trigonometric identities, and the sum of two angles of a triangle, known as compound angles. There are some other compound angle-based identities too for other trigonometric ratios. Based on these compound angle identities, transformation of formulae has been done. These transformations have been done using the sum and products of the trigonometric ratios. This transformation of sum and product identities are stated as “sin(A+B) + sin(A-B) = 2sinAcosB”. Using all these identities, there are some advanced identities too. In order to calculate problems based on multiple angles, there are some specific identities for multiple angles associated with trigonometric ratios. The formulae of multiple angles are “sin2A = 2sinAcosA”, and many more.  There are some other identities too for other trigonometric ratios. In order to solve some major problems related to sub-angles, there are some specific identities associated with these trigonometric ratios. Based on submultiples angles, specific trigonometric identities or formulae have been included. Such as “sinA/2 = 2sinA/2cosA/2”, and many more. These identities have so many advantages that are discussed in next section.

Importance of trigonometric formulae

In order to solve various mathematical and geometrical problems in mathematics, there is so much importance of these identities or formulae. In order to conduct some research work for finding some formula in physics, or in chemistry, these trigonometric formulae are very useful to solve these major problems. There is a huge advantage of these formulae in physics, chemistry, engineering, geology, geography, and in many fields too. Due to solve some huge problems, specific identities are used. These trigonometric formulae are also useful and important to solve calculus-related problems or triangle-related problems. Sometimes, to calculate the position of some satellites, planes, or any planets, these formulae are used to calculate properly. Based on some trajectory, these identities are very useful and efficient to find the solution of some major problem. These formulae are mostly used in the calculus field, to demonstrate, and solve major problems. Generally, the area of some particular region can be calculated by using integration, these identities are very helpful for providing easiness and efficiency to calculate them.

Conclusion

Based on the above discussion, all types of trigonometric identities have been depicted. These identities or formulae are very important to solve some particular problem. Those problems are generally solved by using trigonometric ratios and identities that are discussed in the above sections. These trigonometric ratios and identities are used to solve the height and distance problems. In many engineering sectors, these formulae are used to solve major problems. Such problems that are using triangle-related problems have been solved by these trigonometric identities. These ratios and formulae provide many advantages while solving some major problems or questions.