Question Number 160762 by mathlove last updated on 06/Dec/21
$$\mathrm{sin}\:\mathrm{10}+\mathrm{sin}\:\mathrm{20}+\mathrm{sin}\:\mathrm{30}+\mathrm{sin}\:\mathrm{40}+\centerdot\centerdot\centerdot\centerdot+\mathrm{sin}\:\mathrm{360}=? \\ $$
Commented by som(math1967) last updated on 06/Dec/21
$$\mathrm{0} \\ $$
Commented by mathlove last updated on 06/Dec/21
$${How}\:\:{solution} \\ $$
Commented by som(math1967) last updated on 06/Dec/21
![sinx+sin2x+...+sinnx=((sin(((n+1)/2))xsin((nx)/2))/(sin(x/2))) if x=10 ∴10,20,30...360 36 term ∴n=36 ∴sin10+sin20+...+sin360 =((sin((370)/2)sin((360)/2))/(sin((10)/2))) =((sin185sin180)/(sin5))=0 [∵sin180=0]](https://www.tinkutara.com/question/Q160773.png)
$${sinx}+{sin}\mathrm{2}{x}+…+{sinnx}=\frac{{sin}\left(\frac{{n}+\mathrm{1}}{\mathrm{2}}\right){xsin}\frac{{nx}}{\mathrm{2}}}{{sin}\frac{{x}}{\mathrm{2}}} \\ $$$${if}\:{x}=\mathrm{10}\: \\ $$$$\therefore\mathrm{10},\mathrm{20},\mathrm{30}…\mathrm{360}\:\:\:\:\:\mathrm{36}\:{term} \\ $$$$\therefore{n}=\mathrm{36} \\ $$$$\therefore{sin}\mathrm{10}+{sin}\mathrm{20}+…+{sin}\mathrm{360} \\ $$$$=\frac{{sin}\frac{\mathrm{370}}{\mathrm{2}}{sin}\frac{\mathrm{360}}{\mathrm{2}}}{{sin}\frac{\mathrm{10}}{\mathrm{2}}} \\ $$$$=\frac{{sin}\mathrm{185}{sin}\mathrm{180}}{{sin}\mathrm{5}}=\mathrm{0} \\ $$$$\left[\because{sin}\mathrm{180}=\mathrm{0}\right] \\ $$
Commented by MJS_new last updated on 06/Dec/21
$$\mathrm{sin}\:{A}°\:=−\mathrm{sin}\:\left(\mathrm{180}+{A}\right)° \\ $$$$\mathrm{we}\:\mathrm{don}'\mathrm{t}\:\mathrm{need}\:\mathrm{to}\:\mathrm{know}\:\mathrm{more} \\ $$
Commented by som(math1967) last updated on 06/Dec/21
$${yes}\:{sir} \\ $$$${sin}\mathrm{10}+{sin}\mathrm{20}+…+{sin}\mathrm{180} \\ $$$$+{sin}\mathrm{190}+{sin}\mathrm{200}+…{sin}\mathrm{360} \\ $$$$={sin}\mathrm{10}+{sin}\mathrm{20}+…+{sin}\mathrm{180} \\ $$$$−{sin}\mathrm{10}−{sin}\mathrm{20}…+{sin}\mathrm{360} \\ $$$$=\mathrm{0} \\ $$