Value Of Standard Trigonometric Expressions

Maxima and minima refer to the maximum and minimum values of trigonometric functions when calculated with a certain angle value. There are several ways of doing it, and one of the most practised methods is differentiation. One way of seeing this is by understanding trigonometric expressions’ maximum and minimum values. Let us begin with a little introduction to find and explain the Maxima minima of linear trigonometric expression. In addition to that, we will also be sharing some quick steps to determine the Maxima minima of quadratic, trigonometric expression. Read more here.

Let’s look at the values of trigonometric functions and ratios at different standard angles.

Now that you have become familiar with the standard angle trigonometric functions values let us move forward to understand the methods of finding the Maxima minima of trigonometric functions.

The maximum and minimum values of trigonometric functions

The maximum value for sine and cosine is 1, and both functions’ minimum values are the same as -1. 

As explained above, the maximum and minimum value of sine and cosine is 1 and -1, and secant and cosecant are inverses of cosine and sine, respectively. These two functions have an undefined Maxima and minima. 

Maxima and minima of trigonometric expressions 

Let’s understand the concept of a maximum-minimum of trigonometric functions and expressions. 

Let us suppose there is a linear trigonometric expression, and we have to determine the maximum-minimum of linear trigonometric expression. 

The linear expression is m sinθ + n cosθ 

This linear trigonometric expression has two individual functions: sine and cosine. Now when sin is at its peak value, the cosine becomes zero. It happens similarly with cosine and sine. To determine the maxima and minima of linear trigonometric expression, we must change the expression and transform it into a more suitable trigonometric function with a change coefficient. While transforming and changing the expression, a suitable coefficient with function cosine or sine generates automatically with a different angle base than the original expression. Now that you have become familiar with the idea and concept of changing the coefficient and equation transformation. Let’s understand the process of converting expressions into single-term trigonometric functions.

Guide to transforming the equation

You need to use the compound angle relationship for equation transformation. 

Sin ( A+ B )=sinAcosB+cosAsinB

We need to transform the function given below into a single function.

a sinθ + b cos θ

Let us suppose 

b=c sin a

a=c cos a

Now A2+b2 = C2

If we draw the graph of the above scenario, it will be a right-angled triangle allowing us to determine the tangent trigonometric functions using the formula. 

So, tan a= b÷a

a sinθ + b cosθ

=> c cos a sinθ + c sinθ a cosθ 

c (cos a sinθ + sin a cosθ )

c sin (θ + a)

Guide to find the Maxima minima of linear trigonometric expression with additive and subtractive nature

With our old expression, a sinθ + b cosθ, we will understand the process to find the Maxima minima of linear trigonometric expression.

Additive 

According to the nature of sine and cosine, the Maxima of the expression, a sin + bcos, can be determined as sin(θ + a) has a maximum value of 1. 

So the Maximum value of the expression will be √(A2+b2)

Whereas the minimum value will be -√(A2+b2)

Guide to determine the Maxima minima of quadratic trigonometric expression

Now that you have understood the process to determine the Maxima minima of linear trigonometric expression. Let us move to the Maxima minima of quadratic trigonometric expression. 

Let us assume the quadratic expression.

3 sin2θ + 2 cos2θ 

According to the trigonometric equation sin2θ + cos2θ = 1, it can be determined that the function needs to be transformed into 3 + cos2θ. 

Max [ 3 sin2θ + 2 cos2θ ] 

Max[ 3 + cos2θ ]

3+Max [ cos2θ ]

3+1 = 4

To determine the minima, we will use the minimum value of cos 

Min [ 3+cos2θ ]

3+ Min[ cos2θ ]

3+0=3

This was all about the process to determine the Maxima minima of quadratic trigonometric expression.

Conclusion

The term Maxima refers to the maximum value of any function or expression, and minimum refers to the minimum value of any function or expression. While determining the trigonometric function maximum and minimum value, we need to follow a certain process of equation transformation to operate with only one function involved. This article has discussed and provided information important for equation transformation and coefficient change to determine the maximum-minimum of linear trigonometric expression and quadratic trigonometric expression, the values of trigonometric functions and ratios at different standard angles and also about AM GM inequality.