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“Trigonometry” is that part of mathematics that deals with the “alignment, measurement and arrangement of the angles and lengths of triangles”. “Trigonometric functions” play an important role in solving complex problems related to angles. Different trigonometric functions have different values, therefore it is essential to understand how the values and trigonometric formulas are derived following the basic principles of geometry. The trigonometric identities and values are formulated by analysis of the angles and adjacent sides of the angles. The triangle with a right angle plays a significant role in giving birth to trigonometric identities thereby making a significant impact upon the branch of trigonometry that exclusively deals with angles. All these aspects are presented in the study.
Determination of trigonometric values
As it has been mentioned in the introductory section, there are various types of angles defined on the basis of their measurements. The angles that are commonly used in the study of geometry are the “acute angle”, “obtuse angle”, “right angle”, “reflex angle”, and “straight angle”. Discussing further, it is to be stated that there are different values associated with each type of angle with respect to its measurement. An acute is an angle that has a value less than “90 degrees”, a right part of the angle measures “90 degrees”, “obtuse angle” measures greater than 90 degrees.
The trigonometric identities are based on the trigonometric ratios that are basically derived from the opposite side, hypotenuse, and adjacent sides of a triangle that consists of a right angle. There are six trigonometric ratios such as “sine”, “cosine”, “secant”, “cotangent”, “tangent”, and “cosecant”. The “Pythagorean theorem” can be used in determining the trigonometric identities such as the perpendicular, hypotenuse, and adjacent sides are used to get the trigonometric identities. The “trigonometric functions” are used in algebra, and calculus-related calculations. The basic formulas related to “trigonometric functions” are as follows:
“Sinθ” = “Perpendicular/ Hypotenuse”
“Cosθ” = “Base/ Hypotenuse”
“Tanθ” = “Perpendicular/ Base”
This shows that the trigonometric identities are derived from the trigonometric ratios from a triangle constant of a right angle.
Discussion of the trigonometric functions
The basic trigonometric functions are “tangent”, “cosecant”, “sine”, “cotangent”, “cosine”, and “cosecant”. There are some “reciprocal trigonometric identities” that form an integral part in the formation of trigonometric formulas and identities.
In the case of “Sin θ” the “reciprocal identity” is “1/Cosec θ”. On the other hand, “cosecθ” is equal to “1/Sin θ”, whereas in the case of “Cos θ” it is equal to the value “1/Sec θ”. Considering the value of “Sec θ” with respect to “reciprocal angle” “1/Cos θ” is achieved. In case of, “Tan θ” the value is equal to “1/Cot θ”, whereas “Cot θ” has a reciprocal value such as “1/Tan θ”
The trigonometric identities are based on the “Pythagorean theorem” and are derived from the alignment and arrangement of the adjacent slides, perpendicular hypotenuse of a triangle that contains angles. The basic identities derived from the Pythagorean model are as follows:
“Sin2 a + cos2a” = “1”
“1+tan2 a” = “sec2 a”
“cosec2a” = “1 + cot2 a”
There are certain trigonometric identities that are based on the trigonometric ratios such as:
“Tan θ” = “Sin θ/Cos θ”
“Cot θ” = “Cos θ/Sin θ”
There are some “supplementary angles” that are associated with trigonometric identities. Supplementary angles are angles whose two angles make a sum of “90 degrees”. The measurement of the angles of a triangle containing the right angle is 45, 60, 0, 30, and 90 degrees. The measurements are made by dealing with heh lengths of a triangle. The trigonometric problems can be solved with the help of trigonometric values. “Radians” and the areas of angles are used to determine the trigonometric identities and get the measurements. The trigonometric values play a significant role in solving the trigonometric sums that are widely used to derive the solution to an everyday problem related to angular measurements. The values of the Sinθ, Cotθ, Sec θ, Cosec θ, tan, and cosθ vary from a range between 0 and 90 degrees. For “Sinθ” the value is 0 for 0 degrees, ½ for 30 degrees, 1/√2 for 45 degrees, and √3/2 for 60 degrees, and 1 for 90 degrees. The values for “cos θ” under “0, 30, 45, 60, and 90 degrees” are 1, √3/2, 1/√2, and 0 respectively. In the case of “tanθ” the values under 0, 30, 45, 60, and “90 degrees” are “0”, “1/√3”, “1”, “√3”, and “infinity respectively”. For “Cotθ” the values are infinity, “√3”, “1”, “1/√3”, and “0” for “0”, “30”, “45”, “60”, and “90” respectively. The values for “Secθ” in “90, 30, 0, 60, 45 and degrees” are”1 “,”2/√3”, “√2, 2”, and “infinity respectively”. Finally, the values for cosecθ for “0, 30, 45, 60, and 90 degrees” are “infinity”, “2”, “√2”, “2/√3”, and “1” respectively.
Conclusion
Trigonometric functions have a wide range in the field of mathematics, and the trigonometric values are derived from the basic trigonometric ratios. The Trigonometric identities have their origin from the traditional Pythagoras theorem, and the values are derived through the ratios of cosine, sine, and tangent. The trigonometric functions play a key role in solving some intricate angular measurement problems that cannot be dealt with manually. These formulas contribute largely to the study of astronomy and geography as long distances can be easily measured with the help of trigonometric values and formulas.
