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I’m doing some trig questions from a book to brush up my trig knowledge. And I have come across two questions that I can’t seem to find solutions

First Question

If $A$ and $B$ are acute angles, find $A+B$, given:

(a) $\tan A = 1/4$, $\tan B = 3/5$. Hint: $\tan (A + B) = 1$

(b) $\tan A =5/3$, $\tan B = 4$.

According to the above hint I know $A+B$ must be $45$ degrees. But other than that I don’t know how $\tan A= 1/4$, $\tan B=3/5$ come in to the picture. I would appreciate if anyone can help me understand how to solve this kind of problems.

Second question is

Find the values of $\sin 2A$, $\cos 2A$, and $\tan 2A$, given that $\tan A = u$, in quadrant one

I know how to find $\tan2A$ using identities and answer for that is $2u/(1-u^2)$

But for $\cos2A$ and $\sin2A$, I can’t get the answers given in the book, The answers given in the book are $$ \sin2A = \frac{2u}{1+u^2},\qquad \cos2A = \frac{1-u^2}{1+u^2} $$

Again highly appreciate if anyone can help me out on these.

Thank you

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Use $\tan(x+y) = \frac{sin(x+y)}{cos(x+y)}$, and the sum of angles formula. –  chubakueno Jan 4 at 7:01
    
cud u mabe fiks yur speling punctuashin an capitalizashun? its reel destractin –  dfeuer Jan 4 at 7:06
    
There are nice solutions using complex numbers. –  lhf Jan 4 at 12:13

1 Answer 1

First use $\displaystyle \tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}$

We know if $\displaystyle \tan x=\tan \alpha$

The general value of $x$ is $n\pi+\alpha$ where $n$ is an integer

Here $\alpha=\frac\pi4$

Then use this to find the principal value of $A+B$

For the second, use Weierstrass substitution

share|improve this answer
    
And in the second one, OP is just supposed to manipulate the double agle formulas in a clever way. –  chubakueno Jan 4 at 7:05
    
@chubakueno,please find the edited version. Also, "quadrant one" in the second ques, right? –  lab bhattacharjee Jan 4 at 7:06
    
Yes, I think so. –  chubakueno Jan 4 at 7:13
    
Thank you for your reply. For 1st question what i don't understand is how do you use tanA=1/4, tanB=3/5 to arrive at the answer for (a) and tanA = 5/3 and tanB=4 for (b). the answers given in the book are (a) 45 degrees (b) 135 degrees –  user119020 Jan 4 at 10:40
    
Just put the values of $\tan A,\tan B$ in the formula I mentioned in the first line of the answer –  lab bhattacharjee Jan 4 at 10:42

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