Thread: cos sin

1. cos sin

what is the exact value of 100(cos1+cos2+...cos44)/(sin1+sin2+...sin44)

2. Hello,

Recall the identities :

$\displaystyle \cos(x)+\cos(y)=2 \cos \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right)$

$\displaystyle \sin(x)+\sin(y)=2 \sin \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right)$

Rearrange $\displaystyle \cos 1+\cos 2+\dots+\cos 43+\cos 44=(\cos 1+\cos 44)+(\cos 2+\cos 43)+\dots+(\cos 22+\cos 23)$
Using the above formula, this gives :

$\displaystyle =2 \bigg( \cos \frac{45}{2} \cos \frac{43}{2}+\cos \frac{45}{2} \cos \frac{41}{2} + \dots + \cos \frac{45}{2} \cos \frac 12 \bigg)$
$\displaystyle =2 \cos \frac{45}{2} \bigg(\cos \frac{43}{2}+\cos \frac{41}{2}+ \dots +\cos \frac 12\bigg)$

Similarly, we get :
$\displaystyle \sin 1+\sin 2+\dots+\sin 43+\sin 44=(\sin 1+\sin 44)+(\sin 2+\sin 43)+\dots+(\sin 22+\sin 23)$

$\displaystyle =2 \bigg(\sin \frac{45}{2} \cos \frac{43}{2}+\sin \frac{45}{2} \cos \frac{41}{2} + \dots + \sin \frac{45}{2} \cos \frac 12 \bigg)$
$\displaystyle =2 \sin \frac{45}{2} \bigg(\cos \frac{43}{2}+\cos \frac{41}{2}+ \dots +\cos \frac 12\bigg)$

The expression is now :

$\displaystyle \frac{100(\cos 1+\cos 2+\dots+\cos 44)}{\sin 1+\sin 2+\dots+\sin 44}=100 \cdot \frac{2 \cos \frac{45}{2} \bigg(\cos \frac{43}{2}+\cos \frac{41}{2}+ \dots +\cos \frac 12\bigg)}{2 \sin \frac{45}{2} \bigg(\cos \frac{43}{2}+\cos \frac{41}{2}+ \dots +\cos \frac 12\bigg)}$

$\displaystyle =100 \cdot \frac{\cos \frac{45}{2}}{\sin \frac{45}{2}}=\boxed{100 \cot \frac{45}{2}}$

Brilliant.