Maximum and Minimum Values of Trigonometric Functions for JEE

VSAT 2022

Trigonometric Functions and Their Values

Trigonometric functions are the functions defined on the basis of the angles of a triangle. These functions describe the relationship established between the sides and angles of a triangle. The basic trigonometric functions are sin, cosine, tangent, secant, cosecant, and cotangent. Various trigonometric identities emerge from these basic trigonometric functions and are formed with the help of the relationship of compound angles of the triangles. 


As the trigonometric functions are cyclic, all trigonometric functions are defined in certain intervals, i.e., each of them has its respective domain and range. According to their domain and range, the maximum and minimum values of these trigonometric functions were predetermined. This article also answers questions like how to find maximum and minimum values of a trigonometric function. It also determines the maximum and minimum value of the quadratic equation formula.


What are Mathematical Functions?

Mathematical functions are those functions that establish the relationship between the one independent variable and the dependent variable. These functions are used to describe the relationship which is used for formulating the physical relationship of practical application and procuring their solution. There are various types of mathematical functions such as polynomial function, trigonometric function, exponential function, and complex function.

Basics of Trigonometric Function

As discussed in the first section, the trigonometric function is the cyclic function that is used to describe the relationship between the sides and angles of the triangle. The basic trigonometric functions are defined with the help of a right-angled triangle in the reference to angle A.

Right-Angled


Image : Right-Angled Triangle ABC

 

There are various basic trigonometric functions which are defined below:

  • Sin 

The sin function of an angle is the ratio of the perpendicular to the hypotenuse of the right-angled triangle. The formula for the sin function of angle A is shown below:

$Sin~{C}=\dfrac{\text{perpendicular}}{\text{Hypotenuse}}$

  • Cosine

The cosine function of an angle is the ratio of the base to the hypotenuse of the right-angled triangle. The formula for the cosine function of angle A is shown below:

$Cosine~{C}=\dfrac{\text{Base}}{\text{Hypotenuse}}$

  • Tangent

The tangent function of an angle is the ratio of the perpendicular to the base of the right-angled triangle. The formula for the tangent function of angle A is shown below: 

$Tangent{C}=\dfrac{\text{Perpendicular}}{\text{Base}}$

  • Cotangent

The cotangent function of an angle is the ratio of the base to the perpendicular of the right-angled triangle. It is also known as the reciprocal of the tangent function. The formula for the cotangent function of angle A is shown below:

$Cotangent~{C}=\dfrac{\text{Base}}{\text{Perpendicular}}$

  • Secant

The secant function of an angle is the ratio of the hypotenuse to the base of the right-angled triangle. It is the reciprocal of the cosine function. The formula for the secant function of angle A is shown below: 

$Secant~C=\dfrac{\text{Hypotenuse}}{\text{Base}}$

  • Cosecant

The cosecant function of an angle is the ratio of the hypotenuse to the perpendicular of the right-angled triangle. It is the reciprocal of a sine function. The formula for the cosecant function of angle A is shown below:

$Cosecant~C=\dfrac{\text{Hypotenuse}}{\text{Perpendicular}}$


Determination of Maximum and Minimum Value of Trigonometric Functions

The maximum and minimum values of the basic trigonometric functions depend upon the domain and range of the respective trigonometric functions. The domain belongs to the angles of the trigonometric functions and the range of the trigonometric functions belongs to the values of the trigonometric function on those angles. The maximum and minimum values of trigonometric functions depend upon the range of the trigonometric functions. The maximum and minimum values of the trigonometric function are given below:


Trigonometric Function

Domain

Range

Maximum Values

Minimum Values

Sine

$(-\infty,\infty)$

$[-1,1]$

1

-1

Cosine

$(-\infty,\infty)$

$[-1,1]$

1

-1

Tangent

$R-(2n+1)\dfrac{\pi}{2}$

$(-\infty,\infty)$

$\infty$

$-\infty$

Cotangent

$(R-n\pi)$

$(-\infty,\infty)$

$\infty$

$-\infty$

Secant

$R-(2n+1)\dfrac{\pi}{2}$

$(-\infty,-1]\cup[1,+\infty)$

$\infty$

$-\infty$

Cosecant

$(R-n\pi)$

$(-\infty,-1]\cup[1,+\infty)$

$\infty$

$-\infty$


Quadratic Equation

A quadratic equation is the equation of a second-degree polynomial where it has a minimum of one term which is a higher degree 2. It is one of the types of a polynomial function. In this quadratic equation, the constant ‘a’ cannot be zero. The general form of the quadratic equation is given below:

ax2+bx+c

In the above equation, the equation with one variable having a higher degree 2 is x and a, b, and c are constants. Various methods used for solving the quadratic equation are factorization, using square roots, discriminant formula, and completing the square.


Maximum and Minimum Value of Quadratic Equation Formula

The maximum and minimum values of the quadratic equation will be determined with the help of the quadratic formula which is given below: 

$D=b^2-4ac$

For determining the maximum and minimum value of the quadratic formula, the equation is given as:

$\begin{align} &y=a x^{2}+b x+c \\ &a x^{2}+b x+c-y=0 \end{align}$

For finding the zeros of the above equation by discriminant formula:

$\begin{align} &b^{2}-4 a(c-y) \geq 0 \\ &b^{2}-4 a c+4 a y \geq 0 \\ &4 a y \geq 4 a c-b^{2} \end{align}$                                 …. (1)

For determining the minimum and maximum values of quadratic equation, the value of constant a keeping greater than zero and less than zero 

When (a > 0), using equation (1),

$\begin{align} &4 a y \geq 4 a c-b^{2} \\ &y \geq 4 a c-\frac{b^{2}}{4 a} \end{align}$

It can be seen that whenever the value of constant a is greater than zero, it will give the minimum value of y. Therefore, the minimum value of the quadratic equation is $4ac-\dfrac{b^2}{4a}$. 

When (a < 0), using equation (1),

$\begin{align} &4 a y \geq 4 a c-b^{2} \\ &y \leq 4 a c-\frac{b^{2}}{4 a} \end{align}$

It can be seen that whenever the value of constant a is smaller than zero, it will give the maximum value of y. Therefore, the maximum value of the quadratic equation is $4ac-\dfrac{b^2}{4a}$. 


Conclusion

This article concludes with an understanding of the trigonometric function and quadratic equation. It gives the maximum and minimum values of the trigonometric function. It also shares the determination of the maximum and minimum values of the quadratic equation formula. The entire article has a better understanding of the theoretical concepts of trigonometry and polynomials. It has also answered queries like how to find the maximum and minimum values of trigonometric functions. It also finds out the maximum and minimum value of the quadratic equation formula.

FAQs on Maximum and Minimum Values of Trigonometric Functions for JEE

1. What is the weightage of the trigonometric section in the competitive exam in JEE? Is it important?

The trigonometric section contains 5% weightage in the competitive exam of JEE. This topic is very important for the preparation of JEE. This concept involves the basic information about the sin, cosine, cotangent, secant, tangent, and cosecant angles. From this topic, various trigonometric identities evolved from the four basic identities. This topic requires high input by practising problems on the related topic. Among this topic, the application of trigonometry, the inverse of trigonometry function is also important from the JEE perspective.

2. What makes trigonometric ratios and trigonometric functions different?

Trigonometric functions are real functions that link an angle of a right-angled triangle to ratios of two side lengths. They are also known as circular functions, angle functions, or goniometric functions in Mathematics. Sin, cosine, tangent, and their reciprocals, cosecant, secant, and cotangent, are the six trigonometric ratios in total. Trigonometric functions are real functions with a defined range and domain that link an angle of a right triangle to ratios of two side lengths.

3. What are the applications of trigonometric functions? 

Trigonometry has many applications in oceanography and is used to estimate the height of tides in oceans. The sine and cosine functions are vital to describe the periodic functions of sound and light waves. Trigonometry has instilled its demand in calculus and its theories. The basic use of trigonometry lies in the measurement of the heights of buildings and mountains. In subjects such as Physics, flight engineering, marine engineering, Biology, Mathematics, and many other fields. This is also used in creating maps and navigation.

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