Question

What is the purpose of cotangent, secant and cosecant?

Answer 1

Hint: For solving this question you should know about the trigonometric functions and their uses. As we know that trigonometric functions are very much used in math and these are used to solve very many problems which are subject to mathematics but they are also important for our life. These are generally some formulas but these used at various places in solving the mathematical problems.

Complete step by step solution:
As our question is asking for telling purposes of the cotangent, secant and cosecant.
The cotangent and secant and cosecant all these are the reciprocal of Tan and cos and sin and we can say that,
The values of these functions will also be the same as the reciprocal of the values of tan, cos and sin.
We know that \[{{\sec }^{2}}\theta =1+{{\tan }^{2}}\theta \], this is also a Pythagorean identity and this is used in most of the problems beside those three trig functions. The secant is useful since the derivative of the tangent function is the square of the secant function.
We can say that the practical use of the cotangent is only \[\cot x=\dfrac{\cos x}{\sin x}\] and if we talk about the cos then it is a phase – shifted sin.
Naming Ratios: For 3 numbers there are 6 ratios that can be made. Sine, cosine and Tangent name 3 of the ratios and three you mentioned allow us to name all 6 ratios.
Avoiding division: - Example –
If we know that the angle of elevation on the top of a 12 meter tall building is \[{{40}^{\circ }}\] and we want to find the distance to the building, call it x. We could use \[\dfrac{12}{x}=\tan {{40}^{\circ }}\]. So, \[x=\dfrac{12}{\tan {{40}^{\circ }}}\].
Now, we get a decimal approximation for \[\tan {{40}^{\circ }}\approx 0.8391\] and do the division. OR we use \[\dfrac{x}{12}=\cot {{40}^{\circ }}\]. So, \[x=12\cot {{40}^{\circ }}\].
We get \[\cot {{40}^{\circ }}\approx 1.1518\] and multiply.
Also, in calculus many of us prefer to work with multiplication of functions rather than division and prefer.

Note: During answering this question we should know how cotangent, secant and cosecant changes with their respective functions and it is generally done by taking reciprocal or taking inverse of this. And these are the same with one another at different angles.