Cos product

Geometry Level 5

Evaluate cos ⁡ a ⋅ cos ⁡ 2 a ⋅ cos ⁡ 3 a ⋯ cos ⁡ 999 a \large{\cos a\cdot \cos 2a\cdot\cos 3a\cdots\cos 999 a} where a = 2 π 1999 a=\dfrac{2\pi}{1999} .

If the value of above product is in the form 2 n 2^{n} , find n n .


The answer is -999.

View wiki
Rohit Udaiwal by Rohit Udaiwal , 20, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Raven Herd
Feb 18, 2016
  • Let P = \(cos a cos2a cos3a......cos999a
  • Q = s i n a s i n 2 a s i n 3 a . . . . . . . . . s i n 999 a sina sin2a sin3a.........sin999a
  • 2 9 99 2^999 PQ =( 2 s i n a c o s a 2 sina cos a ) ( 2 s i n 2 a c o s 2 a ) (2sin2a cos2a) .......... ( 2 s i n 999 a c o s 999 a ) (2sin999a cos999a)
  • = ( s i n 2 a s i n 4 a . . . . . . s i n 998 a ) (sin2a sin4a ......sin998a) [ − s i n ( 2 p i − 1000 a ) [-sin(2pi - 1000a) . [ − s i n ( 2 p i − 1002 a ) ] . [-sin(2pi -1002a)]. ... . [ − s i n ( 2 p i − 1998 a ) ] .[-sin(2pi - 1998a)]
  • = s i n 2 a s i n 4 a . . . . . . s i n 998 a s i n 999 a s i n 997 a . . . . . . . . . . s i n a sin2a sin4a ......sin998a sin999a sin997a ..........sin a =Q
    • P = 1 / 2 ( 999 ) P=1/2^(999)
0 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...