Answer
409.8k+ views
Hint: Here, we will expand the given expression as the sum of 100 terms, by substituting the value of \[k\] from 1 to 100 in the summation. We will find the summation in the reverse order and add both the summation. We will then use the trigonometric formula to solve it further and find the required value of the given expression.
Formula Used:
We will use the formula \[\sin \left( {A + B} \right) = \sin A\cos B + \cos A\sin B\].
Complete step-by-step answer:
Let the value of \[S = \sum\limits_{k = 1}^{100} {\sin \left( {kx} \right)\cos \left( {101 - k} \right)x} \].
Now, in this question, the sigma symbol , \[\sum {} \] denotes the sum of multiple terms starting from the value of \[k = 1\] to the value of \[k = 100\].
In \[S = \sum\limits_{k = 1}^{100} {\sin \left( {kx} \right)\cos \left( {101 - k} \right)x} \], if we substitute the values of \[k = 1,2,3,....,100\], then we can write this summation as:
\[S = \sin x\cos 100x + \sin 2x\cos 99x + ....\sin 100x\cos x\]………………….. \[\left( 1 \right)\]
Now, rewriting the equation \[\left( 1 \right)\] by writing it in the reverse order, we get,
\[S = \sin 100x\cos x + \sin 99x\cos 2x + .....\sin x\cos 100x\]…………………. \[\left( 2 \right)\]
Now, adding the equations \[\left( 1 \right)\] and \[\left( 2 \right)\],
\[2S = \left( {\sin x\cos 100x + \sin 100x\cos x} \right) + \left( {\sin 2x\cos 99x + \sin 99x\cos 2x} \right) + ...\left( {\sin 100x\cos x + \sin x\cos 100x} \right)\]
Now, using the formula \[\sin \left( {A + B} \right) = \sin A\cos B + \cos A\sin B\], we get
\[ \Rightarrow 2S = \left[ {\sin \left( {x + 100x} \right) + \sin \left( {2x + 99x} \right) + ..\sin \left( {100x + x} \right)} \right]\]
Adding the terms in the bracket, we get
\[ \Rightarrow 2S = \left[ {\sin \left( {101x} \right) + \sin \left( {101x} \right) + ..\sin \left( {101x} \right)} \right]\]
As the total number of terms are 100, so we get
\[ \Rightarrow 2S = 100\sin \left( {101x} \right)\]
Dividing both sides by 2, we get
\[ \Rightarrow S = 50\sin \left( {101x} \right)\]
But, we have assumed that \[S = \sum\limits_{k = 1}^{100} {\sin \left( {kx} \right)\cos \left( {101 - k} \right)x} \]
Therefore, we can say that,
\[\sum\limits_{k = 1}^{100} {\sin \left( {kx} \right)\cos \left( {101 - k} \right)x} = 50\sin \left( {101x} \right)\]
Hence, the value of \[\sum\limits_{k = 1}^{100} {\sin \left( {kx} \right)\cos \left( {101 - k} \right)x} \] is equal to \[50\sin \left( {101x} \right)\]
Therefore, option C is the correct answer.
Note: A summation means the act of adding or doing a cumulative sum of the given element by substituting the different values of the same variable in the same element and adding them together. Now, the basic difference between a summation and a sigma is that, summation is the adding up of the given series of elements whereas, a sigma is just a mathematical symbol used to indicate this summation without stating anything. Hence, summation plays an important role for finding out the aggregate value of a given element from its lower limit to upper limit of summation.
Formula Used:
We will use the formula \[\sin \left( {A + B} \right) = \sin A\cos B + \cos A\sin B\].
Complete step-by-step answer:
Let the value of \[S = \sum\limits_{k = 1}^{100} {\sin \left( {kx} \right)\cos \left( {101 - k} \right)x} \].
Now, in this question, the sigma symbol , \[\sum {} \] denotes the sum of multiple terms starting from the value of \[k = 1\] to the value of \[k = 100\].
In \[S = \sum\limits_{k = 1}^{100} {\sin \left( {kx} \right)\cos \left( {101 - k} \right)x} \], if we substitute the values of \[k = 1,2,3,....,100\], then we can write this summation as:
\[S = \sin x\cos 100x + \sin 2x\cos 99x + ....\sin 100x\cos x\]………………….. \[\left( 1 \right)\]
Now, rewriting the equation \[\left( 1 \right)\] by writing it in the reverse order, we get,
\[S = \sin 100x\cos x + \sin 99x\cos 2x + .....\sin x\cos 100x\]…………………. \[\left( 2 \right)\]
Now, adding the equations \[\left( 1 \right)\] and \[\left( 2 \right)\],
\[2S = \left( {\sin x\cos 100x + \sin 100x\cos x} \right) + \left( {\sin 2x\cos 99x + \sin 99x\cos 2x} \right) + ...\left( {\sin 100x\cos x + \sin x\cos 100x} \right)\]
Now, using the formula \[\sin \left( {A + B} \right) = \sin A\cos B + \cos A\sin B\], we get
\[ \Rightarrow 2S = \left[ {\sin \left( {x + 100x} \right) + \sin \left( {2x + 99x} \right) + ..\sin \left( {100x + x} \right)} \right]\]
Adding the terms in the bracket, we get
\[ \Rightarrow 2S = \left[ {\sin \left( {101x} \right) + \sin \left( {101x} \right) + ..\sin \left( {101x} \right)} \right]\]
As the total number of terms are 100, so we get
\[ \Rightarrow 2S = 100\sin \left( {101x} \right)\]
Dividing both sides by 2, we get
\[ \Rightarrow S = 50\sin \left( {101x} \right)\]
But, we have assumed that \[S = \sum\limits_{k = 1}^{100} {\sin \left( {kx} \right)\cos \left( {101 - k} \right)x} \]
Therefore, we can say that,
\[\sum\limits_{k = 1}^{100} {\sin \left( {kx} \right)\cos \left( {101 - k} \right)x} = 50\sin \left( {101x} \right)\]
Hence, the value of \[\sum\limits_{k = 1}^{100} {\sin \left( {kx} \right)\cos \left( {101 - k} \right)x} \] is equal to \[50\sin \left( {101x} \right)\]
Therefore, option C is the correct answer.
Note: A summation means the act of adding or doing a cumulative sum of the given element by substituting the different values of the same variable in the same element and adding them together. Now, the basic difference between a summation and a sigma is that, summation is the adding up of the given series of elements whereas, a sigma is just a mathematical symbol used to indicate this summation without stating anything. Hence, summation plays an important role for finding out the aggregate value of a given element from its lower limit to upper limit of summation.
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
When people say No pun intended what does that mea class 8 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Name the states which share their boundary with Indias class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give an account of the Northern Plains of India class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Advantages and disadvantages of science
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Bimbisara was the founder of dynasty A Nanda B Haryanka class 6 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
10 examples of evaporation in daily life with explanations
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
How do you graph the function fx 4x class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)