# Solve the equation : $2\cos 22{\dfrac{1}{2}^0}\sin 22{\dfrac{1}{2}^0}\,\,$ by using trigonometric formulas or identities.

Last updated date: 21st Mar 2023

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Answer

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Hint - In order to solve this problem use the formula that $\sin 2\theta = 2\sin \theta \cos \theta $. Then put the value of angle.

As we know the formula,

$\sin 2\theta = 2\sin \theta \cos \theta $ ……(i)

And the given equation is $2\cos 22{\dfrac{1}{2}^0}\sin 22{\dfrac{1}{2}^0}\,\,$

We also know $a\dfrac{b}{c} = \dfrac{{ac + b}}{c}$ ……(ii)

From (ii) we can say, $22\dfrac{1}{2} = \dfrac{{22(2) + 1}}{2} = \dfrac{{44 + 1}}{2} = \dfrac{{45}}{2}$ ……(iii)

Therefore the given equation can be written as ,

$2\cos {\dfrac{{45}}{2}^0}\sin {\dfrac{{45}}{2}^0}\,\,$

And from (i) we can say,

$2\cos {\dfrac{{45}}{2}^0}\sin {\dfrac{{45}}{2}^0}\,\, = \sin \dfrac{{2\,{\text{x }}45}}{2} = \sin {45^0}$

We know the value of sin45 is $\dfrac{1}{{\sqrt 2 }}$.

Therefore from the above equations we can say

$2\cos 22{\dfrac{1}{2}^0}\sin 22{\dfrac{1}{2}^0} = \dfrac{1}{{\sqrt 2 }}$.

Hence the answer to this question is $\dfrac{1}{{\sqrt 2 }}$.

Note – In these types of problems of trigonometry we have to use the general formula of trigonometry, after observing which formula can be fit into the given question then solve the equation according to the formula. There is an alternative method to solve this question , it is , if we know the values of the angle given in the equation we can also directly put it and get the actual value of the equation.

As we know the formula,

$\sin 2\theta = 2\sin \theta \cos \theta $ ……(i)

And the given equation is $2\cos 22{\dfrac{1}{2}^0}\sin 22{\dfrac{1}{2}^0}\,\,$

We also know $a\dfrac{b}{c} = \dfrac{{ac + b}}{c}$ ……(ii)

From (ii) we can say, $22\dfrac{1}{2} = \dfrac{{22(2) + 1}}{2} = \dfrac{{44 + 1}}{2} = \dfrac{{45}}{2}$ ……(iii)

Therefore the given equation can be written as ,

$2\cos {\dfrac{{45}}{2}^0}\sin {\dfrac{{45}}{2}^0}\,\,$

And from (i) we can say,

$2\cos {\dfrac{{45}}{2}^0}\sin {\dfrac{{45}}{2}^0}\,\, = \sin \dfrac{{2\,{\text{x }}45}}{2} = \sin {45^0}$

We know the value of sin45 is $\dfrac{1}{{\sqrt 2 }}$.

Therefore from the above equations we can say

$2\cos 22{\dfrac{1}{2}^0}\sin 22{\dfrac{1}{2}^0} = \dfrac{1}{{\sqrt 2 }}$.

Hence the answer to this question is $\dfrac{1}{{\sqrt 2 }}$.

Note – In these types of problems of trigonometry we have to use the general formula of trigonometry, after observing which formula can be fit into the given question then solve the equation according to the formula. There is an alternative method to solve this question , it is , if we know the values of the angle given in the equation we can also directly put it and get the actual value of the equation.

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