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In trigonometry, there are in total six trigonometric functions, which are as follows- sine, cosine, and tangent are major functions, while secant, Cosecant, and cot are the secondary functions. Moreover, the angle functions come in a variety of shapes and sizes. These six elementary functions utilised and used in trigonometry are sin, cos, Cosecant sec, tangent, and cotangent.
The reciprocal functions of sine, cosine, and tangent functions, respectively, are Cosecant, secant, and cotangent. The cosecant function is the reciprocal of the sine function and is one of the six trigonometric functions available.
The Cosecant of an angle in a right triangle is obtained by dividing the length of the hypotenuse by the length of the opp in a right angle triangle.sides. It is basically the reciprocal of sin. So cosecant X is equal to 1/ sine X.
For Example-
If the value of Sine Y is equal to 4/7. Then the cosecant Y will be 1/ sine Y. Which in furtherance will be 1 divided by 4/7=7/4. So, cosecant Y will be 7/4
If the value of Sine Y is equal to 3/5, then the value of Cosecant Y will be 1/ sine Y which is equal to 5/3
If the value of Sine Y is equal to 5/8 then the value of cosec Y will be 1 divided by Sine Y which is 8/5
Table of Cosec value in different Angles
As we know, cosec Y is equal to 1 divided by sine Y, where Y denotes the angle of the triangle.
Inverse of cosec function
Every trigonometric function, such as the cosec has an inverse function which has the working in the other direction. All of the inverse functions have the similar name as the original but add the word ‘arc’ to the beginning. As a result, the inverse of cosec is arccosec, and so on. “Angles whose is A,” we deduce when we see “arccosec A.”
Cosec 30 is equal to 2.00 which further denotes that the csc of 30° will be 2.000.
Arccosec 2 is equal to 30 which further denote: an angle with a cosecant of 2.0 equals 30 degrees.
Sometimes abbreviated as arccosec or cosec^-1.
Cosec where the angle is greater than 90 Degrees
The two variable angles in this i.e. (the int. angles of a triangle) of a right triangle are always less than 90°. We can still calculate the cosecant of any angle, no matter how large it is, as well as the cosecant of negative angles.
Derivative of cosec Y
The derivative of cosec Y is -cosec Y cot Y, which is the negative of the product of the trigonometric functions cosec Y and cot Y. The process of assessing the derivative of cosec Y with respect to angle Y is called the differentiation of cosec Y.
So, d(cosec y)/dy = (cosec y)’ = -cot y cosec y is the differentiation of cosec y with regard to angle y. The derivative of sine y can be used to determine the derivative of cosec y. Differentiation of cosec y can be accomplished in a variety of methods. The limit definition, chain rule, and quotient rule can all be used to find the derivative of cosec y.
Subsequently, the derivation of the cosec y formula will be written as the following-
d(cosec y)/dy = -cot y cosec y
(cosec y)’ = -cot y cosec y
The derivative of csc y cot y can be known with the help of the rule of the product as we have known that d(cosec y)/dy = -cot y csc y, d(cot y)/dy = -csc2y
So by the product rule, we have
d(cosec y cot y)/dy = (csc y cot y)’
(cosec y)’ cot y + cosec y (cot y)’
= -cosec y cot y cot y + cosec y (-cosec2y)
= -cosec y (cot2y + cosec2y)
Therefore, the derivative of cosec y cot y is -csc y (cot2y + csc2y).
Consequently, for example to find the second derivation of the cosecant y then we have known that the first derivative of cosec y is -cosec y cot y.
To find out the second derivative of the cosec y, we differentiate -cosec y cot y using the rule of product as followed-
So, by the help of product rule, we can take-
(cosec y)” = (-cosec y cot y)’
(-cosec y)’ cot y + (-cosec y) (cot y)’
= cosec y cot y cot y + (-cosec y) (-cosec2y)
= cosec y (cot2y + cosec2y)
Therefore, in this situation the second derivative of cosec y is cosec y (cot2y + cosec2y).
Conclusion
Trigonometric functions are quite relevant and important in a variety of situations. Many fields use these functions to work with distance connections based on a right triangle; the term “triangulation” also comes from this. Trigonometric functions are practical and valuable in forensic research, criminal investigations, and video games, to name a few.
