
The minimum value of is
A)
B)
C)
D)
Answer
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Hint:
Here, we will first convert all these functions in terms of the sine and cosine. Then we will add them together and simplify it further to find the minimum value of the sum. Trigonometric ratios are defined as the ratios of any two sides of a right angled triangle.
Formula Used:
We will use the following Formula:
1) Trigonometric ratio:
2) Trigonometric identities:
3) The square of the sum of two numbers is given by the formula:
4) The difference of square of two numbers is given by the algebraic identity
5) Trigonometric formula:
6)
Complete step by step solution:
We are given with a trigonometric function
Let
Now, we have to represent all the terms in terms of sine and cosine.
Now, substituting and in the given expression, we get
By grouping the terms and taking L.C.M., we get
Adding the like terms, we get
Now, by using the trigonometric identity , we get
………………………..
Using the algebraic identity , we can write
Rewriting the equation, we get
Substituting in equation , we get
Substituting in the above equation, we get
The difference of square of two numbers is given by the algebraic identity
………………………..
We know that .
Multiplying and dividing by on both the sides, we get
Now, by using trigonometric formula , we get
Thus , by this identity, we will get .
Substituting
Now, by using the property , we get
Now, considering in this case the value is positive.
We know that Arithmetic mean Geometric mean. Thus, we will get
Thus, we get
which is positive…………………………………………………..
Now, considering in this case the value is negative
We know that Arithmetic mean Geometric mean. Thus, we will get
Thus, we get
which is positive………………………………………………
Since the value has to be minimum, from and , we get
Thus the minimum value is
Therefore, the minimum value of is .
Note:
We can find the limits by using the trigonometric identity and values. So, it becomes important to remember all the basic identities and values. We have found the minimum value using the relation between arithmetic mean and geometric mean. Arithmetic mean or average is defined as the sum of numbers divided by the quantity of numbers, Geometric mean of two numbers is defined as the square root of their product. We should also remember that the Arithmetic mean should never be less than the geometric mean.
Here, we will first convert all these functions in terms of the sine and cosine. Then we will add them together and simplify it further to find the minimum value of the sum. Trigonometric ratios are defined as the ratios of any two sides of a right angled triangle.
Formula Used:
We will use the following Formula:
1) Trigonometric ratio:
2) Trigonometric identities:
3) The square of the sum of two numbers is given by the formula:
4) The difference of square of two numbers is given by the algebraic identity
5) Trigonometric formula:
6)
Complete step by step solution:
We are given with a trigonometric function
Let
Now, we have to represent all the terms in terms of sine and cosine.
Now, substituting
By grouping the terms and taking L.C.M., we get
Adding the like terms, we get
Now, by using the trigonometric identity
Using the algebraic identity
Rewriting the equation, we get
Substituting
Substituting
The difference of square of two numbers is given by the algebraic identity
We know that
Multiplying and dividing by
Now, by using trigonometric formula
Thus , by this identity, we will get
Substituting
Now, by using the property
Now, considering
We know that Arithmetic mean
Thus, we get
Now, considering
We know that Arithmetic mean
Thus, we get
Since the value has to be minimum, from
Thus the minimum value is
Therefore, the minimum value of
Note:
We can find the limits by using the trigonometric identity and values. So, it becomes important to remember all the basic identities and values. We have found the minimum value using the relation between arithmetic mean and geometric mean. Arithmetic mean or average is defined as the sum of numbers divided by the quantity of numbers, Geometric mean of two numbers is defined as the square root of their product. We should also remember that the Arithmetic mean should never be less than the geometric mean.
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