Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 195571 by York12 last updated on 05/Aug/23

$$letf(x+y)+f(x-y)=2f\left(x\right)f\left(y\right)\wedge f\left(\frac{1}{2}\right)=-1$$$$compute\underset{k=1}{\sum ^{20}}\left[\frac{1}{\sin \left(k\right)\sin (k+f\left(k\right))}\right]$$

Answered by mahdipoor last updated on 05/Aug/23

$$x=1/2y=0\Rightarrow 2f\left(1/2\right)=2f\left(1/2\right)f\left(0\right)\Rightarrow f\left(0\right)=1$$$$x=1/2y=1/2\Rightarrow f\left(1\right)+f\left(0\right)=2f^2\left(1/2\right)\Rightarrow f\left(1\right)=1$$$$x=1y=1\Rightarrow f\left(2\right)+f\left(0\right)=2f^2\left(1\right)\Rightarrow f\left(2\right)=1$$$$x=2y=1\Rightarrow f\left(3\right)+f\left(1\right)=2f\left(2\right)f\left(1\right)\Rightarrow f\left(3\right)=1$$$$....$$$$x=ny=1\Rightarrow f\left(n\right)+f(n-1)=2f\left(n\right)f\left(1\right)$$$$\Rightarrow f\left(1\right)=f(n-1)=1\Rightarrow f\left(n\right)=1$$$$....$$$$\mathrm{\Sigma }=\underset{1}{\sum ^{20}}\frac{1}{sin\left(k\right)sin(k+1)}$$

Answered by mr W last updated on 05/Aug/23

$$f\left(x\right)=\cos \left(ax\right)$$$$f\left(\frac{1}{2}\right)=\cos \left(\frac{a}{2}\right)=-1\Rightarrow \frac{a}{2}=\pi$$$$\Rightarrow f\left(x\right)=\cos \left(2\pi x\right)$$$$forx\in Z,f\left(x\right)=1$$$$\underset{k=1}{\sum ^{20}}\frac{1}{\sin \left(k\right)\sin (k+f\left(k\right))}$$$$=\underset{k=1}{\sum ^{20}}\frac{1}{\sin \left(k\right)\sin (k+1)}$$$$\approx 1.54106054$$

Commented by York12 last updated on 05/Aug/23

$$$$$$\Rightarrow f\left(x\right)=cos\left(2\pi x\right)$$$$\Rightarrow \underset{k=1}{\sum ^{20}}\left[\frac{1}{\sin \left(k\right)\sin (k+f\left(k\right))}\right]=\underset{k=1}{\sum ^{20}}\left[\frac{1}{sin\left(k\right)sin(k+1)}\right]$$$$=cosec\left(1\right)\underset{k=1}{\sum ^{20}}\left[\frac{sin\left(1\right)}{sin\left(k\right)sin(k+1)}\right]=cosec\left(1\right)\underset{k=1}{\sum ^{20}}\left[\frac{sin(k+1-k)}{sin\left(k\right)sin(k+1)}\right]$$$$=cosec\left(1\right)\underset{k=1}{\sum ^{20}}\left[\frac{sin(k+1)cos\left(k\right)-cos(k+1)sin\left(k\right)}{sin\left(k\right)sin(k+1)}\right]$$$$=cosec\left(1\right)\underset{k=1}{\sum ^{20}}[cot\left(k\right)-cot(k+1)]$$$$=cosec\left(1\right)[cot\left(1\right)-cot\left(21\right)]\approx 3133.3715$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com