$\cot x = 1/\tan x$

Calculus Level 5

$V = 2 \sin(2^\circ) + 4 \sin(4^\circ) + 6 \sin(6^\circ) + \cdots + 180 \sin (180^\circ)$ .

For $V$ as defined above, find $\lfloor V \rfloor$ .

1 solution

Ossama Ismail
Feb 27, 2017

$V = 2 \sin 2 + 4 \sin 4 + \cdots + 178 \sin 178$

$\sin(1) V = 2 \sin(1) \sin(2) + 4 \sin(1) \sin(4) + \cdots + 178 \sin(1) \sin(178)$

We know that $2 \sin (a) \sin (b) = \cos(a - b) - \cos(a + b)$ , then

$\sin(1) V = 2 \sin(1) \sin(2) + 2 (2 \sin(1) \sin(4) + \cdots + 89( 2 \sin(1) \sin(178)$

$\sin(1) V = \cos(1) - \cos(3) + 2 ( \cos(3) - \cos(5) ) + \cdots + 89( \cos(177) - \cos(179))$

$\sin(1) V = \cos(1) + \cos(3) + \cos(5) ) + \cdots + \cos (175) + \cos (177) - 89 \cos(179)$

$\sin(1) V = \cos(1) + (\cos(3) + \cos(177) ) + \cdots + ( \cos (89) + \cos (91)) - 89 \cos(179)$

$\sin(1) V = \cos (1) + 89 \cos ( 1 ) = 90 \cos( 1)$

$V = 90 \cos(1)/\sin(1) = 90 \cot(1) = 90 \times 57.289961 = 5156.0965$

$\lfloor V \rfloor = 5156$

Why have you treated that sum as only over even x?

Brian Moehring - 4 years, 3 months ago

Yes ....i forgot to mention that

Ossama Ismail - 4 years, 3 months ago

Put a backslash " \ " in front of functions to make them proper (non-italic) $\sin$ , $\cos$ , and $\cot$ .

Chew-Seong Cheong - 4 years, 3 months ago

thanks >>> done

Ossama Ismail - 4 years, 3 months ago