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if tan(a-b)=1/4, tan(a+b)=1/3 then tan2a=?

 Oct 14, 2014

Best Answer 

 #2
avatar+95369 
+10

$$\\atan(\frac{1}{4})=A-B\qquad (1)\\\\
atan(\frac{1}{3})=A+B\qquad(2)\\\\
(1)+(2)\\\\
atan(\frac{1}{4})+atan(\frac{1}{3})=2A\\\\
tan[atan(\frac{1}{4})+atan(\frac{1}{3})]=tan[2A]\\\\$$

 

 

$$$Now I am going to use the identity$\\\\
\boxed{tan(\theta_1+\theta_2)=\frac{tan(\theta_1)+tan(\theta_2)}{1-tan(\theta_1)tan(\theta_2)}}\\\\\\
tan(2A)\\\\
=tan[atan(\frac{1}{4})+atan(\frac{1}{3})]\\\\
=\frac{\frac{1}{4}+\frac{1}{3}}{1-\frac{1}{4}\times\frac{1}{3}}\\\\
=\frac{\frac{7}{12}}{1-\frac{1}{12}}\\\\
=\frac{7}{12}\div \frac{11}{12}\\\\
=\frac{7}{12}\times \frac{12}{11}\\\\
=\frac{7}{11}\\\\
$This answer is exact$$$

.
 Oct 14, 2014
 #1
avatar+94619 
+5

Using the tangent inverse

tan-1(1/4) = (a - b) = about 14.04°  ......  and tan-1(1/3) = (a +b) = about 18.43°

So

(a -b) + (a+b) = (14.04 + 18.43)° →  

2a = about 32.47°

So

tan(2a) = tan(32.47°) = about .636363= 7/11

 

 Oct 14, 2014
 #2
avatar+95369 
+10
Best Answer

$$\\atan(\frac{1}{4})=A-B\qquad (1)\\\\
atan(\frac{1}{3})=A+B\qquad(2)\\\\
(1)+(2)\\\\
atan(\frac{1}{4})+atan(\frac{1}{3})=2A\\\\
tan[atan(\frac{1}{4})+atan(\frac{1}{3})]=tan[2A]\\\\$$

 

 

$$$Now I am going to use the identity$\\\\
\boxed{tan(\theta_1+\theta_2)=\frac{tan(\theta_1)+tan(\theta_2)}{1-tan(\theta_1)tan(\theta_2)}}\\\\\\
tan(2A)\\\\
=tan[atan(\frac{1}{4})+atan(\frac{1}{3})]\\\\
=\frac{\frac{1}{4}+\frac{1}{3}}{1-\frac{1}{4}\times\frac{1}{3}}\\\\
=\frac{\frac{7}{12}}{1-\frac{1}{12}}\\\\
=\frac{7}{12}\div \frac{11}{12}\\\\
=\frac{7}{12}\times \frac{12}{11}\\\\
=\frac{7}{11}\\\\
$This answer is exact$$$

Melody Oct 14, 2014
 #3
avatar+95369 
+5

Thanks Chris    

 Oct 14, 2014
 #4
avatar+94619 
+5

I did mine "on the fly"....Melody's approach is actually "better"

 

 Oct 14, 2014

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