Answer
Verified
430.2k+ views
Hint: We know a formula $\sin \left( {A + B} \right) = \sin A\cos B + \cos A\sin B$ and here, we have to evaluate the value of $\sin {36^ \circ }\cos {9^ \circ } + \cos {36^ \circ }\sin {9^ \circ }$ which resembles with the right hand side of the above written formula. So, we can apply the above given formula to find the required value of the above given question.
Complete step-by-step solution:
Here, the given expression is $\sin {36^ \circ }\cos {9^ \circ } + \cos {36^ \circ }\sin {9^ \circ }$.
We know a formula $\sin \left( {A + B} \right) = \sin A\cos B + \cos A\sin B$.
And when we compare the right hand side of the above given formula with the given expression in above question. We get, $A = 3{6^ \circ }$ and $B = {9^ \circ }$.
So, by applying above formula we can write $\sin {36^ \circ }\cos {9^ \circ } + \cos {36^ \circ }\sin {9^ \circ }$ as $\sin \left( {{{36}^ \circ } + {9^ \circ }} \right)$.
$ \Rightarrow \sin {36^ \circ }\cos {9^ \circ } + \cos {36^ \circ }\sin {9^ \circ } = \sin \left( {{{36}^ \circ } + {9^ \circ }} \right)$
$ \Rightarrow \sin {36^ \circ }\cos {9^ \circ } + \cos {36^ \circ }\sin {9^ \circ } = \sin {45^ \circ }$
By using the trigonometry table we can find the value of $\sin {45^ \circ } = \dfrac{1}{{\sqrt 2 }}$.
So, the value of $\sin {36^ \circ }\cos {9^ \circ } + \cos {36^ \circ }\sin {9^ \circ }$is $\dfrac{1}{{\sqrt 2 }}$.
Thus, option (B) is correct.
Note: Similarly, some important formulae which may be used to solve similar types of problems.
(1) $\sin \left( {A - B} \right) = \sin A\cos B - \cos B\sin A$ .
(2) $\cos \left( {A + B} \right) = \cos A\cos B - \sin A\sin B$.
(3) $\cos \left( {A - B} \right) = \cos A\cos B + \sin A\sin B$.
While applying these formulae firstly we have to make sure that the given expression must resemble the right side of the above given formulae. After this we have to find the value of $\cos ine$ and $\sin $ of some angles and that value can be found by using a trigonometry table.
If the above problem is modified as $\sin {36^ \circ }\cos {9^ \circ } - \cos {36^ \circ }\sin {9^ \circ }$ then we have to apply the first formula given in the hint section.
Similarly, we can apply a second formula when we have to evaluate the value of mathematical expressions like $\cos {36^ \circ }\cos {9^ \circ } - \sin {36^ \circ }\sin {9^ \circ }$.
Similarly, we can apply a third formula when we have to evaluate the value of mathematical expressions like $\cos {36^ \circ }\cos {9^ \circ } + \sin {36^ \circ }\sin {9^ \circ }$.
Complete step-by-step solution:
Here, the given expression is $\sin {36^ \circ }\cos {9^ \circ } + \cos {36^ \circ }\sin {9^ \circ }$.
We know a formula $\sin \left( {A + B} \right) = \sin A\cos B + \cos A\sin B$.
And when we compare the right hand side of the above given formula with the given expression in above question. We get, $A = 3{6^ \circ }$ and $B = {9^ \circ }$.
So, by applying above formula we can write $\sin {36^ \circ }\cos {9^ \circ } + \cos {36^ \circ }\sin {9^ \circ }$ as $\sin \left( {{{36}^ \circ } + {9^ \circ }} \right)$.
$ \Rightarrow \sin {36^ \circ }\cos {9^ \circ } + \cos {36^ \circ }\sin {9^ \circ } = \sin \left( {{{36}^ \circ } + {9^ \circ }} \right)$
$ \Rightarrow \sin {36^ \circ }\cos {9^ \circ } + \cos {36^ \circ }\sin {9^ \circ } = \sin {45^ \circ }$
By using the trigonometry table we can find the value of $\sin {45^ \circ } = \dfrac{1}{{\sqrt 2 }}$.
So, the value of $\sin {36^ \circ }\cos {9^ \circ } + \cos {36^ \circ }\sin {9^ \circ }$is $\dfrac{1}{{\sqrt 2 }}$.
Thus, option (B) is correct.
Note: Similarly, some important formulae which may be used to solve similar types of problems.
(1) $\sin \left( {A - B} \right) = \sin A\cos B - \cos B\sin A$ .
(2) $\cos \left( {A + B} \right) = \cos A\cos B - \sin A\sin B$.
(3) $\cos \left( {A - B} \right) = \cos A\cos B + \sin A\sin B$.
While applying these formulae firstly we have to make sure that the given expression must resemble the right side of the above given formulae. After this we have to find the value of $\cos ine$ and $\sin $ of some angles and that value can be found by using a trigonometry table.
If the above problem is modified as $\sin {36^ \circ }\cos {9^ \circ } - \cos {36^ \circ }\sin {9^ \circ }$ then we have to apply the first formula given in the hint section.
Similarly, we can apply a second formula when we have to evaluate the value of mathematical expressions like $\cos {36^ \circ }\cos {9^ \circ } - \sin {36^ \circ }\sin {9^ \circ }$.
Similarly, we can apply a third formula when we have to evaluate the value of mathematical expressions like $\cos {36^ \circ }\cos {9^ \circ } + \sin {36^ \circ }\sin {9^ \circ }$.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Which are the Top 10 Largest Countries of the World?
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Difference Between Plant Cell and Animal Cell
Give 10 examples for herbs , shrubs , climbers , creepers
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you graph the function fx 4x class 9 maths CBSE
Write a letter to the principal requesting him to grant class 10 english CBSE