1-Value-of-20-21-1-22-2-60-40-is-2-Sum-of-all-solutions-of-eq-n-cos-3-sin-2-in-interval-pi-2-pi-2-is-

Question Number 58406 by rahul 19 last updated on 22/Apr/19

1) V a l u e o f 20! + \frac{21!}{1!} + \frac{22!}{2!} + \dots . + \frac{60!}{40!} i s ?

2) S u m o f a l l s o l u t i o n s o f {e q}^{n} :

\cos 3 θ = \sin 2 θ i n i n t e r v a l [- \frac{π}{2}, \frac{π}{2}] i s ?

Answered by MJS last updated on 23/Apr/19

(1)

29 623 769 613 544 887 392 514 571 468 800 000

Commented by rahul 19 last updated on 23/Apr/19

S i r, k i n d l y s h o w y o u r w o r k i n g \dots

A n s \to 20!^{61} C_{21} .

Commented by MJS last updated on 23/Apr/19

to be honest, I just computed it \dots

Commented by rahul 19 last updated on 23/Apr/19

h o w m u c h t i m e d i d i t t a k e ?

Answered by MJS last updated on 23/Apr/19

(2)

\cos 3 θ = \sin 2 θ

\cos θ (1 - 2 \sin θ) (1 + 2 \sin θ) = 2 \cos θ \sin θ

\cos θ = 0 \Rightarrow θ = - \frac{π}{2} \lor θ = \frac{π}{2}

{4 \sin}^{2} θ - 2 \sin θ - 1 = 0

\sin θ = - \frac{1}{4} \pm \frac{\sqrt{5}}{4} \Rightarrow θ = - \frac{3 π}{10} \lor θ = \frac{π}{10}

\Rightarrow answer is - \frac{π}{5}

Commented by rahul 19 last updated on 23/Apr/19

t h a n k s s i r!