Values of Trigonometric Ratios

The word trigonometry has been derived from the Greek word trigonon meaning triangle and metron which means to measure. Trigonometry is considered the most ancient subject among all. It has been studied by several scholars around the world. The ratio of the side length of the triangle is known as the trigonometric ratio. The ratios are related to ratio sides of the right triangle to their respective angle three basic ratios of trigonometry are sine, cosine and tangent. The other significant ratios are cosec sec and cot which are respectively derived from sine, cosine and tan.

Definition of trigonometric ratios

Six trigonometric ratios are there in trigonometry such as sine, cosine, tangent, secant, cosecant and cotangent.  The values of these ratios are based on the angle of the right-angled triangle. If a right-angled triangle is being considered, the edges of the triangle are as A, B and C. Therefore the sides can be stated as AB, BC and CA. CA is considered as the hypotenuse of the triangle. BC is the base and AB is the perpendicular. The angle is denoted as d. The ratios of trigonometry are defined as:

Sine: the angle sine can be defined as a ratio of perpendicular to hypotenuse, thus for the given example of triangle sine d = AB/AC.

Cosine: the angle cosine can be defined as the ratio of base to hypotenuse, thus for the given example of triangle cos d = BC/AC.

Tangent: the angle of tan can be defined as the ratio of perpendicular to base, thus for the given example of triangle tan d = AB/BC

Cosecant: the angle of cosec can be defined as the ratio of hypotenuse to perpendicular, thus for the given example of triangle cosec d = AC/AB

Secant: the angle of sec can be defined as the ratio of hypotenuse to base, thus for the given example of triangle sec d = AC/BC

Cotangent: the angle of cotangent can be defined as the ratio of base to perpendicular, thus for the given example of triangle cot d = BC/AB

Formulas in trigonometric ratios

Basic formulation of the ratios has been mentioned in the above section. But several other formulas are there that can be used as an alternative with respect to specific situations. Some of the formulas are mentioned below:

“Sin d = 1/cosec d”

“Cos d = 1/sec d”

“Tan d = 1/cot d”

“Cosec d = 1/sin d”

“Sec d = 1/cos d”

“Cot d = 1/tan d”

Complementary angle trigonometric ratios have some identities. The sum of two complementary angles is 90 degrees. Therefore the angle complementing it is (90° – d). The complementary angle ratios are:

“Sin (90° – d) = cos d”

“Cos (90° – d) = sin d”

“Cosec (90° – d) = sec d”

“Sec (90° – d) = cosec d”

“Tan (90° – d) = cot d”

“Cot (90° – d) = tan d”

Some Pythagorean trigonometric ratio identities are there. These formulas have been derived from Pythagorus theorem. Three such formulas are:

“Sin2d + cos2d = 1”

“1+ tan2d = sec2d”

“1 + cot2d = cosec2d”

“sin (A + B) = cos A sin B + sin A cos B”

 “cos (A + B) = cos A cos B – sin A sin B”

“tan (A + B) = (tan A + tan B)/ (1 – tan A tan B)”

Application of trigonometric ratios

Trigonometric formulas are applicable in several areas. The fields where trigonometric ratios are used are as follows: astronomy, satellite systems, naval industry, aviation, oceanography, land survey and cartography. The main usage of trigonometry is accurately measuring distances. Several direct applications of trigonometry are often observed in daily life. While building up the mechanical structure of bridges or building applications of trigonometric values are being noticed. The preparation of the blueprint of the plan includes several trigonometric calculations based on which necessary measuring calculations are done. Measurement of unknown sides and angles can be determined easily. Preparing the draft of a plan is very important in construction therefore modern architecture is specifically dependent on trigonometry. In many sectors elevation angles are calculated through trigonometric ratios.

 Trigonometric properties are applicable in measuring distance, height and surface area. Music production has also witnessed the application of trigonometry. Identities such as sine and cosine by implementing basic laws while conducting sound waves. It can be considered as a silent guest during measurement of sound waves by conductors and producers. In the technology of computing, imaging trigonometry plays an important role. Complex and detailed images are generated. Triangulation is the technology of making such images. Measuring the height of mountains is another important application. The concept of shadows and trigonometry are widely used in measuring such vast heights. 

Conclusion

Trigonometry is also applicable in other fields for several functions such as electronics, chemistry, medical imaging, biology, number theory, metrology, cryptology, phonetics, compression, crystallography, cartography, computer graphics and game development. Values of trigonometric ratios are an important educational concept in secondary standards as it enhances conceptual understanding, problem-solving ability and memorisation. It provides a better understanding of the world as several natural structures of earth resemble triangles. It has been applied for right-angled triangles but can also be used for other triangle types. Angles determining trigonometric ratio helps in better determination of derivations and values. Several identities are there that are used in practical application.