if-tanA-tanB-P-and-AtanB-q-express-the-value-of-cos-2A-3B-in-terms-of-p-and-q-

Question Number 33969 by mondodotto@gmail.com last updated on 28/Apr/18

if tanA + tanB = P and AtanB = q

express the value of \cos (2 A + 3 B) in terms of p and q .

Answered by tanmay.chaudhury50@gmail.com last updated on 28/Apr/18

l e t t a n A = a t a n B = b

c o s (2 A + 2 B + B)

= c o s (2 A + 2 B) . c o s B - s i n (2 A + 2 B) . s i n B

= {c o s 2 A . c o s 2 B - s i n 2 A . s i n 2 B} c o s B - {s i n 2 A .

c o s 2 B + c o s 2 A . s i n 2 B} . s i n B

p u t c o s 2 A = (1 - a^{2} / 1 + a^{2}) s i n 2 A = 2 a / 1 + a^{2}

s i m p l y f y i f p r l b l e m c o m m e n t p l s . .

Commented by mondodotto@gmail.com last updated on 29/Apr/18

i dont undstnd

Commented by tanmay.chaudhury50@gmail.com last updated on 29/Apr/18

o k i a m s o l v i n g i n d e t a i l s a n d s h a l l u p l o a d t h e i m a g e

Answered by tanmay.chaudhury50@gmail.com last updated on 29/Apr/18

l e t t a n A = a, t a n B = b s o a + b = p, a b = q

{(a - b)}^{2} = {(a + b)}^{2} - 4 a b

a - b = \sqrt{p^{2} - 4 q} a n d a + b = p

s o l v i n g a = p + \sqrt{p^{2} - 4 q}

b = p - \sqrt{p^{2} - 4 q}

c o s (2 A + 3 B)

= c o s 2 A . c o s 3 B - s i n 2 A . s i n 3 B

= (\frac{1 - {t a n}^{2} A}{1 + {t a n}^{2} A}) c o s B (3 - 4 {c o s}^{2} B) - (\frac{2 t a n A}{1 + {t a n}^{2} A}) . s i n B .

(4 {s i n}^{2} B - 3)

n o w r e p l a c e t a n A b y a t a n B b y b

c o s B = \frac{1}{\sqrt{1 + b^{2}}} s i n B = \frac{t a n B}{s e c B} = \frac{b}{\sqrt{1 + b^{2}}}

\frac{1 - a^{2}}{1 + a^{2}} . \frac{1}{\sqrt{1 + b^{2}}} (3 - \frac{4}{1 + b^{2}}) - (\frac{2 a}{1 + a^{2}}) \frac{b}{\sqrt{1 + b^{2}}} . (\frac{4 b^{2}}{1 + b^{2}} - 3

p u t a = p + \sqrt{p^{2} - 4 q} a n d b = p - \sqrt{p^{2} 4 q}

s i m p l i f y