Question

Find the value of A and B where \[A\& B\] are acute angles. If \[\tan (A + B) = \sqrt 3 \] and \[\tan (A - B) = \dfrac{1}{{\sqrt 3 }}\]

Answer 1

Hint: Write the values of \[\dfrac{1}{{\sqrt 3 }}\& \sqrt 3 \] in terms of tangent then compare it with the angles given in tangent in the question use it to formulate two equation and then solve it to get the value of A and B.
Complete step-by-step answer:
We know that
\[\begin{array}{l}
\tan {60^ \circ } = \sqrt 3 \\
\tan {30^ \circ } = \dfrac{1}{{\sqrt 3 }}
\end{array}\]
So from here we can say that
\[\begin{array}{l}
\tan {60^ \circ } = \sqrt 3 = \tan (A + B)\\
\tan {30^ \circ } = \dfrac{1}{{\sqrt 3 }} = \tan (A - B)
\end{array}\]
 Which means that
\[\begin{array}{l}
\tan {60^ \circ } = \tan (A + B)\\
\tan {30^ \circ } = \tan (A - B)
\end{array}\]
Therefore if we remove tan from both the ides we will get it as
\[\begin{array}{l}
{60^ \circ } = (A + B)...............................(i)\\
{30^ \circ } = (A - B)...............................(ii)
\end{array}\]
Now if we just add both of these equations we will get it as
\[\begin{array}{l}
 \Rightarrow A + B + A - B = {60^ \circ } + {30^ \circ }\\
 \Rightarrow 2A = {90^ \circ }\\
 \Rightarrow A = \dfrac{{{{90}^ \circ }}}{2}\\
 \Rightarrow A = {45^ \circ }
\end{array}\]
Now as we have the value of A let us put it in equation (i) to get the value of B
\[\begin{array}{l}
 \Rightarrow {60^ \circ } = (A + B)\\
 \Rightarrow {45^ \circ } + B = {60^ \circ }\\
 \Rightarrow B = {60^ \circ } - {45^ \circ }\\
 \Rightarrow B = {15^ \circ }
\end{array}\]
As we can see that both of these are acute angles and \[A = {45^ \circ }\& B = {15^ \circ }\]

Note: Acute angles are the angles which are less than \[{{{90}^ \circ }}\] also note that i have used elimination method to solve the equation it can also be easily done by using substitution method, where you can get the value of either A or B from any one of the equation and then put it in the other one so that we can get an equation with only one variable and the continue with the same steps as i have done in the solution.