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Question: Find the value of \(x\) if \(\sin 2x = \sin {60^ \circ }\cos {30^ \circ } - \sin {30^ \circ }\cos {6...

Find the value of xx if sin⁡2x=sin⁡60∘cos⁡30∘−sin⁡30∘cos⁡60∘\sin 2x = \sin {60^ \circ }\cos {30^ \circ } - \sin {30^ \circ }\cos {60^ \circ }
A.20∘{20^ \circ }
B.15∘{15^ \circ }
C.30∘{30^ \circ }
D.45∘{45^ \circ }

Explanation

Solution

Hint : Use the trigonometric identity sin⁡acos⁡b−cos⁡asin⁡b=sin⁡(a−b)\sin a\cos b - \cos a\sin b = \sin (a - b).

Given,
sin⁡2x=sin⁡60∘cos⁡30∘−sin⁡30∘cos⁡60∘\sin 2x = \sin {60^ \circ }\cos {30^ \circ } - \sin {30^ \circ }\cos {60^ \circ } …..(i)
As we know
sin⁡acos⁡b−cos⁡asin⁡b=sin⁡(a−b)\sin a\cos b - \cos a\sin b = \sin (a - b)
When we put a=60,b=30a = 60,b = 30 we get the above equation after assigning value as,
sin⁡60∘cos⁡30∘−sin⁡30∘cos⁡60∘=sin⁡(60∘−30∘)=sin⁡30∘\sin {60^ \circ }\cos {30^ \circ } - \sin {30^ \circ }\cos {60^ \circ } = \sin ({60^ \circ } - {30^ \circ }) = \sin {30^ \circ } …(ii)
From (i) & (ii) We get,
sin⁡30∘=sin⁡2x\sin {30^ \circ } = \sin 2x
2x=30∘ x=15∘  2x = {30^ \circ } \\\ x = {15^ \circ } \\\
Hence the correct option is B.

Note :- In these types of questions we have to apply the basic identities of trigonometry and solve the asked question. We can also assign values of the angles and find the asked value.