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Question Number 57127 by ANTARES VY last updated on 30/Mar/19

{cos1°}+{cos2°}+{cos3°}+....+{cos270}=?

$$\left\{\boldsymbol{\mathrm{cos}}\mathrm{1}°\right\}+\left\{\boldsymbol{\mathrm{cos}}\mathrm{2}°\right\}+\left\{\boldsymbol{\mathrm{cos}}\mathrm{3}°\right\}+....+\left\{\boldsymbol{\mathrm{cos}}\mathrm{270}\right\}=? \\ $$

Commented by tanmay.chaudhury50@gmail.com last updated on 30/Mar/19

source S.L Loney great mathematician

$$\boldsymbol{{source}}\:\boldsymbol{{S}}.\boldsymbol{{L}}\:\boldsymbol{{Loney}}\:\boldsymbol{{great}}\:\boldsymbol{{mathematician}} \\ $$

Commented by Tawa1 last updated on 30/Mar/19

from the general formular:  compare with....     cos(α) + cos(2α) + cos(3α) + ... + cos(nα)                                                                 =  ((cos[(((n + 1)/2))α] sin(((nα)/2)))/(sin((α/2))))  α  =  1,    n = 270     cos(1) + cos(2) + cos(3) + ... + cos(270)                                                                 =  ((cos[(((270 + 1)/2))] sin(((270)/2)))/(sin((1/2))))     cos(1) + cos(2) + cos(3) + ... + cos(270)                                                                 =  ((cos[(135.5)] sin(135))/(sin(0.5)))     cos(1) + cos(2) + cos(3) + ... + cos(270)   =  − 57.7943

$$\mathrm{from}\:\mathrm{the}\:\mathrm{general}\:\mathrm{formular}: \\ $$$$\mathrm{compare}\:\mathrm{with}.... \\ $$$$\:\:\:\mathrm{cos}\left(\alpha\right)\:+\:\mathrm{cos}\left(\mathrm{2}\alpha\right)\:+\:\mathrm{cos}\left(\mathrm{3}\alpha\right)\:+\:...\:+\:\mathrm{cos}\left(\mathrm{n}\alpha\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\:\frac{\mathrm{cos}\left[\left(\frac{\mathrm{n}\:+\:\mathrm{1}}{\mathrm{2}}\right)\alpha\right]\:\mathrm{sin}\left(\frac{\mathrm{n}\alpha}{\mathrm{2}}\right)}{\mathrm{sin}\left(\frac{\alpha}{\mathrm{2}}\right)} \\ $$$$\alpha\:\:=\:\:\mathrm{1},\:\:\:\:\mathrm{n}\:=\:\mathrm{270} \\ $$$$\:\:\:\mathrm{cos}\left(\mathrm{1}\right)\:+\:\mathrm{cos}\left(\mathrm{2}\right)\:+\:\mathrm{cos}\left(\mathrm{3}\right)\:+\:...\:+\:\mathrm{cos}\left(\mathrm{270}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\:\frac{\mathrm{cos}\left[\left(\frac{\mathrm{270}\:+\:\mathrm{1}}{\mathrm{2}}\right)\right]\:\mathrm{sin}\left(\frac{\mathrm{270}}{\mathrm{2}}\right)}{\mathrm{sin}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)} \\ $$$$\:\:\:\mathrm{cos}\left(\mathrm{1}\right)\:+\:\mathrm{cos}\left(\mathrm{2}\right)\:+\:\mathrm{cos}\left(\mathrm{3}\right)\:+\:...\:+\:\mathrm{cos}\left(\mathrm{270}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\:\frac{\mathrm{cos}\left[\left(\mathrm{135}.\mathrm{5}\right)\right]\:\mathrm{sin}\left(\mathrm{135}\right)}{\mathrm{sin}\left(\mathrm{0}.\mathrm{5}\right)} \\ $$$$\:\:\:\mathrm{cos}\left(\mathrm{1}\right)\:+\:\mathrm{cos}\left(\mathrm{2}\right)\:+\:\mathrm{cos}\left(\mathrm{3}\right)\:+\:...\:+\:\mathrm{cos}\left(\mathrm{270}\right)\:\:\:=\:\:−\:\mathrm{57}.\mathrm{7943} \\ $$

Commented by ANTARES VY last updated on 30/Mar/19

?

$$? \\ $$

Commented by tanmay.chaudhury50@gmail.com last updated on 30/Mar/19

convert degree[to radian  →1^o =(π/(180))radian...

$${convert}\:{degree}\left[{to}\:{radian}\:\:\rightarrow\mathrm{1}^{{o}} =\frac{\pi}{\mathrm{180}}{radian}...\right. \\ $$

Commented by tanmay.chaudhury50@gmail.com last updated on 30/Mar/19

Commented by tanmay.chaudhury50@gmail.com last updated on 30/Mar/19

Commented by tanmay.chaudhury50@gmail.com last updated on 30/Mar/19

Commented by tanmay.chaudhury50@gmail.com last updated on 30/Mar/19

Commented by tanmay.chaudhury50@gmail.com last updated on 30/Mar/19

Commented by tanmay.chaudhury50@gmail.com last updated on 30/Mar/19

Commented by peter frank last updated on 30/Mar/19

thank you for sharing

$${thank}\:{you}\:{for}\:{sharing} \\ $$

Commented by malwaan last updated on 31/Mar/19

can you send link to  dowload the book ?  please

$$\boldsymbol{{can}}\:\boldsymbol{{you}}\:\boldsymbol{{send}}\:\boldsymbol{{link}}\:\boldsymbol{{to}} \\ $$$$\boldsymbol{{dowload}}\:\boldsymbol{{the}}\:\boldsymbol{{book}}\:? \\ $$$$\boldsymbol{{please}} \\ $$

Commented by ANTARES VY last updated on 31/Mar/19

yes

$$\boldsymbol{\mathrm{yes}} \\ $$

Commented by ANTARES VY last updated on 31/Mar/19

do you have  a  math  group or  telegramm?

$$\boldsymbol{\mathrm{do}}\:\boldsymbol{\mathrm{you}}\:\boldsymbol{\mathrm{have}}\:\:\boldsymbol{\mathrm{a}}\:\:\boldsymbol{\mathrm{math}} \\ $$$$\boldsymbol{\mathrm{group}}\:\boldsymbol{\mathrm{or}}\:\:\boldsymbol{\mathrm{telegramm}}? \\ $$

Commented by tanmay.chaudhury50@gmail.com last updated on 31/Mar/19

Commented by tanmay.chaudhury50@gmail.com last updated on 31/Mar/19

Commented by tanmay.chaudhury50@gmail.com last updated on 01/Apr/19

Commented by Tawa1 last updated on 01/Apr/19

Sir, please give me hint on how to download good textbook  on archive.org

$$\mathrm{Sir},\:\mathrm{please}\:\mathrm{give}\:\mathrm{me}\:\mathrm{hint}\:\mathrm{on}\:\mathrm{how}\:\mathrm{to}\:\mathrm{download}\:\mathrm{good}\:\mathrm{textbook} \\ $$$$\mathrm{on}\:\mathrm{archive}.\mathrm{org} \\ $$

Commented by tanmay.chaudhury50@gmail.com last updated on 01/Apr/19

click on book then scroll down  apply down load option

$${click}\:{on}\:{book}\:{then}\:{scroll}\:{down}\:\:{apply}\:{down}\:{load}\:{option} \\ $$

Commented by tanmay.chaudhury50@gmail.com last updated on 01/Apr/19

Commented by Tawa1 last updated on 01/Apr/19

What will i search for or how can i search for a particular   textbook sir.  Sorry for disturbing sir.

$$\mathrm{What}\:\mathrm{will}\:\mathrm{i}\:\mathrm{search}\:\mathrm{for}\:\mathrm{or}\:\mathrm{how}\:\mathrm{can}\:\mathrm{i}\:\mathrm{search}\:\mathrm{for}\:\mathrm{a}\:\mathrm{particular}\: \\ $$$$\mathrm{textbook}\:\mathrm{sir}.\:\:\mathrm{Sorry}\:\mathrm{for}\:\mathrm{disturbing}\:\mathrm{sir}. \\ $$

Commented by ANTARES VY last updated on 01/Apr/19

adress

$$\boldsymbol{\mathrm{adress}} \\ $$

Commented by tanmay.chaudhury50@gmail.com last updated on 01/Apr/19

you sesrch topic wise such as algebra or Trigonometry  orcalculus or graph or coordinate geometry  or physics..mechanics  or heat  or electricity

$${you}\:{sesrch}\:{topic}\:{wise}\:{such}\:{as}\:{algebra}\:{or}\:{Trigonometry} \\ $$$${orcalculus}\:{or}\:{graph}\:{or}\:{coordinate}\:{geometry} \\ $$$${or}\:{physics}..{mechanics}\:\:{or}\:{heat}\:\:{or}\:{electricity} \\ $$

Commented by Tawa1 last updated on 01/Apr/19

God bless you sir

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$

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