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.