Solve the equation : $2\cos 22{\dfrac{1}{2}^0}\sin 22{\dfrac{1}{2}^0}\,\,$ by using trigonometric formulas or identities.
Answer
364.8k+ views
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.
Last updated date: 02nd Oct 2023
•
Total views: 364.8k
•
Views today: 6.64k
Recently Updated Pages
What do you mean by public facilities

Difference between hardware and software

Disadvantages of Advertising

10 Advantages and Disadvantages of Plastic

What do you mean by Endemic Species

What is the Botanical Name of Dog , Cat , Turmeric , Mushroom , Palm

Trending doubts
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

Why are resources distributed unequally over the e class 7 social science CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Briefly mention the contribution of TH Morgan in g class 12 biology CBSE

What is the past tense of read class 10 english CBSE
