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Question: Evaluate the value of following: \(\sin {36^ \circ }\cos {9^ \circ } + \cos {36^ \circ }\sin {9^ \...

Evaluate the value of following:
sin⁡36∘cos⁡9∘+cos⁡36∘sin⁡9∘\sin {36^ \circ }\cos {9^ \circ } + \cos {36^ \circ }\sin {9^ \circ }
(A) −12\dfrac{{ - 1}}{{\sqrt 2 }}
(B) 12\dfrac{1}{{\sqrt 2 }}
(C) 2\sqrt 2
(D) −2- \sqrt 2

Explanation

Solution

We know a formula sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin \left( {A + B} \right) = \sin A\cos B + \cos A\sin B and here, we have to evaluate the value of sin⁡36∘cos⁡9∘+cos⁡36∘sin⁡9∘\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∘cos⁡9∘+cos⁡36∘sin⁡9∘\sin {36^ \circ }\cos {9^ \circ } + \cos {36^ \circ }\sin {9^ \circ }.
We know a formula sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\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=36∘A = 3{6^ \circ } and B=9∘B = {9^ \circ }.
So, by applying above formula we can write sin⁡36∘cos⁡9∘+cos⁡36∘sin⁡9∘\sin {36^ \circ }\cos {9^ \circ } + \cos {36^ \circ }\sin {9^ \circ } as sin⁡(36∘+9∘)\sin \left( {{{36}^ \circ } + {9^ \circ }} \right).
⇒sin⁡36∘cos⁡9∘+cos⁡36∘sin⁡9∘=sin⁡(36∘+9∘)\Rightarrow \sin {36^ \circ }\cos {9^ \circ } + \cos {36^ \circ }\sin {9^ \circ } = \sin \left( {{{36}^ \circ } + {9^ \circ }} \right)
⇒sin⁡36∘cos⁡9∘+cos⁡36∘sin⁡9∘=sin⁡45∘\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∘=12\sin {45^ \circ } = \dfrac{1}{{\sqrt 2 }}.
So, the value of sin⁡36∘cos⁡9∘+cos⁡36∘sin⁡9∘\sin {36^ \circ }\cos {9^ \circ } + \cos {36^ \circ }\sin {9^ \circ }is 12\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⁡(A−B)=sin⁡Acos⁡B−cos⁡Bsin⁡A\sin \left( {A - B} \right) = \sin A\cos B - \cos B\sin A .
(2) cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos \left( {A + B} \right) = \cos A\cos B - \sin A\sin B.
(3) cos⁡(A−B)=cos⁡Acos⁡B+sin⁡Asin⁡B\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\cos ine and sin⁡\sin of some angles and that value can be found by using a trigonometry table.
If the above problem is modified as sin⁡36∘cos⁡9∘−cos⁡36∘sin⁡9∘\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∘cos⁡9∘−sin⁡36∘sin⁡9∘\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∘cos⁡9∘+sin⁡36∘sin⁡9∘\cos {36^ \circ }\cos {9^ \circ } + \sin {36^ \circ }\sin {9^ \circ }.