# If $tan\ (A + B) = \sqrt3$ and $tan\ (A - B) = \frac{1}{\sqrt3}$; $0^o < A + B ≤ 90^o; A > B$, find $A$ and $B$.

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Given:

$tan\ (A + B) = \sqrt3$ and $tan\ (A - B) = \frac{1}{\sqrt3}$; $0^o < A + B ≤ 90^o; A > B$

To do:

We have to find $A$ and $B$.

Solution:

$\tan (A-B)=\frac{1}{\sqrt3}$

$\tan (A-B)=\tan 30^{\circ}$          (Since $\tan 30^{\circ}=\frac{1}{\sqrt3}$)

$\Rightarrow A-B=30^{\circ}$......(i)

$\tan (A+B)=\sqrt3$

$\tan (A+B)=\tan 60^{\circ}$             (Since $\tan 60^{\circ}=\sqrt3$)

$\Rightarrow A+B=60^{\circ}$

$\Rightarrow A=60^{\circ}-B$........(ii)

Substituting (ii) in (i), we get,

$60^{\circ}-B-B=30^{\circ}$

$\Rightarrow 2B=30^{\circ}$

$\Rightarrow B=\frac{30^{\circ}}{2}$

$\Rightarrow B=15^{\circ}$

$\Rightarrow A=60^{\circ}-15^{\circ}=45^{\circ}$

The values of $A$ and $B$ are $45^{\circ}$ and $15^{\circ}$ respectively.

Updated on 10-Oct-2022 13:22:12