Answer
Verified
377.4k+ views
Hint: As we can see that in the above question we have to find the value of $ \sin \theta $ which means that there is use of trigonometric application and identities to get the required answer. We will first draw the diagram according to the data given in the question . We will assume one of the angles as $ \phi $ as it cannot be $ \theta $ because it is the angle subtended at the top of the tower by the line joining the points.
Complete step-by-step answer:
Let us first draw the diagram according to the question:
In the above figure we have $ \angle BAD = \theta $ ( $ \theta $ is the angle subtended at the top of the tower by the line joining these points).
Let us assume $ \angle CBA = \phi ,\angle ADC = 90 - \phi $ . Also we have $ CD = a $ and $ BC = b $ .
Now in triangle CAD , we have $ \tan \phi $ , so we have $ \tan \phi = \dfrac{p}{b} $ , here the perpendicular is AC and the base is $ BC = b $ . So by putting the value we can write $ \tan \phi = \dfrac{{AC}}{b} $ .
Again in triangle CAD we can write $ \tan (90 - \phi ) = \dfrac{{AC}}{a} $ .
Now we know that $ \tan (90 - \theta ) = \cot \theta $ , so it can be written as $ \cot \phi = \dfrac{{AC}}{a} $ .
We know the trigonometric identity that $ \cot \theta = \dfrac{1}{{\tan \theta }} $ , so by this we can write from the above expressions , $ \dfrac{{AC}}{a} = \dfrac{1}{{\dfrac{{AC}}{b}}} \Rightarrow \dfrac{{AC}}{a} = \dfrac{a}{{AC}} $ .
By cross multiplication it can be written as $ A{C^2} = ab $ or $ AC = \sqrt {ab} $ .
Now we can write $ \sin \phi = \dfrac{p}{h} $ , here we have perpendicular is $ AC = \sqrt {ab} $ and hypotenuse $ AB $ can be written as $ A{C^2} + B{C^2} $ , we have $ AC = \sqrt {ab} $ and $ BC = b $ .
So we have hypotenuse \[{h^2} = {(\sqrt {ab} )^2} + {b^2}\], by removing the square of hypotenuse we have i.e. $ h = \sqrt {ab + {b^2}} $ .
After putting these values back we have $ \sin \phi = \sqrt {\dfrac{{ab}}{{ab + {b^2}}}} $ .
By squaring both the sides we have $ {\sin ^2}\phi = {\left( {\sqrt {\dfrac{{ab}}{{ab + {b^2}}}} } \right)^2} $ . Since squaring the square root removes it, we have $ \dfrac{{ab}}{{ab + {b^2}}} $ . We can take the common factor out in the denominator, so we have $ \dfrac{{ab}}{{b(a + b)}} = \dfrac{a}{{a + b}} $ .
From the above figure and by applying trigonometric identity we can write $ \sin \theta = - 1 + 2{\sin ^2}\phi $ .
By putting the value in the identity we have $ - 1 + \dfrac{{2a}}{{a + b}} $ , On solving we have $ \dfrac{{ - a - b + 2a}}{{a + b}} \Rightarrow \dfrac{{a - b}}{{a + b}} $ .
Hence the correct option is (b) $ \dfrac{{a - b}}{{a + b}} $ .
So, the correct answer is “Option B”.
Note: We should note that in the above solution we have used the trigonometric identity that $ \sin \theta = - \sin \left( {90 - \phi } \right) = - \cos 2\phi $ . Then there is another identity which says that $ \cos 2\theta $ can be written as $ 1 - 2{\sin ^2}\theta $ , so we can write here $ - \cos 2\phi = - 1 + 2{\sin ^2}\phi $ . In triangle CAD we have perpendicular $ = AC $ and base $ = a $ .
Complete step-by-step answer:
Let us first draw the diagram according to the question:
In the above figure we have $ \angle BAD = \theta $ ( $ \theta $ is the angle subtended at the top of the tower by the line joining these points).
Let us assume $ \angle CBA = \phi ,\angle ADC = 90 - \phi $ . Also we have $ CD = a $ and $ BC = b $ .
Now in triangle CAD , we have $ \tan \phi $ , so we have $ \tan \phi = \dfrac{p}{b} $ , here the perpendicular is AC and the base is $ BC = b $ . So by putting the value we can write $ \tan \phi = \dfrac{{AC}}{b} $ .
Again in triangle CAD we can write $ \tan (90 - \phi ) = \dfrac{{AC}}{a} $ .
Now we know that $ \tan (90 - \theta ) = \cot \theta $ , so it can be written as $ \cot \phi = \dfrac{{AC}}{a} $ .
We know the trigonometric identity that $ \cot \theta = \dfrac{1}{{\tan \theta }} $ , so by this we can write from the above expressions , $ \dfrac{{AC}}{a} = \dfrac{1}{{\dfrac{{AC}}{b}}} \Rightarrow \dfrac{{AC}}{a} = \dfrac{a}{{AC}} $ .
By cross multiplication it can be written as $ A{C^2} = ab $ or $ AC = \sqrt {ab} $ .
Now we can write $ \sin \phi = \dfrac{p}{h} $ , here we have perpendicular is $ AC = \sqrt {ab} $ and hypotenuse $ AB $ can be written as $ A{C^2} + B{C^2} $ , we have $ AC = \sqrt {ab} $ and $ BC = b $ .
So we have hypotenuse \[{h^2} = {(\sqrt {ab} )^2} + {b^2}\], by removing the square of hypotenuse we have i.e. $ h = \sqrt {ab + {b^2}} $ .
After putting these values back we have $ \sin \phi = \sqrt {\dfrac{{ab}}{{ab + {b^2}}}} $ .
By squaring both the sides we have $ {\sin ^2}\phi = {\left( {\sqrt {\dfrac{{ab}}{{ab + {b^2}}}} } \right)^2} $ . Since squaring the square root removes it, we have $ \dfrac{{ab}}{{ab + {b^2}}} $ . We can take the common factor out in the denominator, so we have $ \dfrac{{ab}}{{b(a + b)}} = \dfrac{a}{{a + b}} $ .
From the above figure and by applying trigonometric identity we can write $ \sin \theta = - 1 + 2{\sin ^2}\phi $ .
By putting the value in the identity we have $ - 1 + \dfrac{{2a}}{{a + b}} $ , On solving we have $ \dfrac{{ - a - b + 2a}}{{a + b}} \Rightarrow \dfrac{{a - b}}{{a + b}} $ .
Hence the correct option is (b) $ \dfrac{{a - b}}{{a + b}} $ .
So, the correct answer is “Option B”.
Note: We should note that in the above solution we have used the trigonometric identity that $ \sin \theta = - \sin \left( {90 - \phi } \right) = - \cos 2\phi $ . Then there is another identity which says that $ \cos 2\theta $ can be written as $ 1 - 2{\sin ^2}\theta $ , so we can write here $ - \cos 2\phi = - 1 + 2{\sin ^2}\phi $ . In triangle CAD we have perpendicular $ = AC $ and base $ = a $ .
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
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE