Question

If 0<A<B<π/4 and sin (A+B)=24/25 and cos (A-B)=4/5, then find the value of tan 2A

Answer 1

Answer:

The value of tan(2A) is \frac{3}{4}[/tex]

Step-by-step explanation:

Given $sin(A+B)=\frac{24}{25}$

$cos(A-B)=\frac{4}{5}$

we have to find tan2A

If $sin(A+B)=\frac{24}{25}$ then perpendicular is 24 and hypotenuse is 25 then

By Pythagoras Theorem third side can be calculated as

$25^{2}=24^{2}+Base^2$

⇒ Base is 7

Hence, $tan(A+B)=\frac{24}{7}$

Similarly,

if $cos(A-B)=\frac{4}{5}$ then base is 4 and hypotenuse is 5 then

By Pythagoras Theorem third side can be calculated as

$5^{2}=4^{2}+Perpendicular^2$

⇒ Perpendicular is 3

Hence, $tan(A-B)=\frac{3}{4}$

∵ B>A, then A-B is negative.hence, taken negative

$tan(A-B)=\frac{-3}{4}$

$tan(2A)=tan((A+B)+(A-B))= \frac{tan(A+B)+tan(A-B)}{1-tan(A+B)\times tan(A-B)}$

⇒ $tan(2A)=\frac{\frac{24}{7}+(\frac{-3}{4})}{1-\frac{24}{7}\times (\frac{-3}{4})}$

                      =$\frac{\frac{75}{28}}{\frac{100}{28}}=\frac{3}{4}$