Question

The values of $ \tan \left( {A - B} \right) $ , $ \sec \left( {A + B} \right) $ are $ 1,\dfrac{2}{{\sqrt 3 }} $ . Choose the correct option for the smallest positive value of $ B $ .
1) $ \dfrac{{25}}{{24}}\pi $
2) $ \dfrac{{19}}{{24}}\pi $
3) $ \dfrac{{13}}{{24}}\pi $
4) $ \dfrac{{11}}{{24}}\pi $

Answer 1

Hint: First we need to solve the given ratios to get some equations. After getting the equations we solve for $ B $ . We got a lot of possibilities for $ B $ , but we had asked for the smallest positive possible value for $ B $ . So, from the values we got we check the smallest possible positive value for the answer. That is our final answer.

Complete step-by-step answer:
Given that,
 $ \tan \left( {A - B} \right) = 1 $ , $ \sec \left( {A + B} \right) = \dfrac{2}{{\sqrt 3 }} $
Let us solve these ratios to get the equations we need.
We know that,
 $ \tan \left( {\dfrac{\pi }{4}} \right) = 1 $ , $ \sec \left( {\dfrac{\pi }{6}} \right) = \dfrac{2}{{\sqrt 3 }} $ ,
 $ \Rightarrow \tan \left( {A - B} \right) = \tan \left( {\dfrac{\pi }{4}} \right),\sec \left( {A + B} \right) = \sec \left( {\dfrac{\pi }{6}} \right) $ ,
We know that the periods of $ \tan ,\sec $ are $ \pi ,2\pi $ respectively.
So, the general solutions for the above trigonometric ratios are $ n\pi \pm \dfrac{\pi }{4},2n\pi \pm \dfrac{\pi }{6} $ .
That means,
 $ A - B = n\pi \pm \dfrac{\pi }{4},A + B = 2n\pi \pm \dfrac{\pi }{6} $ ,
Now by the above equations, we can easily for $ B $ , that is by subtracting the first equation from the second equation.
So, by subtracting the first equation from the second equation, we get
 $ 2B = n\pi \pm \dfrac{\pi }{6} \mp \dfrac{\pi }{4} $ ,
 $ \Rightarrow B = \dfrac{{n\pi }}{2} \pm \dfrac{\pi }{{12}} \mp \dfrac{\pi }{8} $ .
So, now we want the smallest possible positive value for $ B $ so put $ n = 1 $ for that.
 $ \Rightarrow B = \dfrac{\pi }{2} \pm \dfrac{\pi }{{12}} \mp \dfrac{\pi }{8} $ ,
We get two values for $ B $ when $ n = 1 $ , those are
 $ B = \dfrac{{11\pi }}{{24}},\dfrac{{13\pi }}{{24}} $
But we need the smallest possible positive value for $ B $ .
So, the required value is
 $ B = \dfrac{{11\pi }}{{24}} $ .
So, the correct option is 4.
So, the correct answer is “Option 4”.

Note: This is a problem where we can make some mistakes. We have to be very careful while solving for $ B $ . It is a place where a lot of students make a mistake because of not changing the sign that is $ \pm $ to $ \mp $ . After getting the answer it is suggestable to check your answer during the exam because of the errors we make.