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Question-219279

Question Number 219279 by Mingma last updated on 22/Apr/25 Answered by som(math1967) last updated on 22/Apr/25 $$\:\frac{\mathrm{2}{sin}\frac{\beta+\alpha}{\mathrm{2}}{cos}\frac{\alpha−\beta}{\mathrm{2}}}{\mathrm{2sin}\:\frac{\alpha+\beta}{\mathrm{2}}{sin}\frac{\alpha−\beta}{\mathrm{2}}}=\sqrt{\mathrm{3}} \\ $$$$\Rightarrow\:{cot}\frac{\alpha−\beta}{\mathrm{2}}={cot}\frac{\pi}{\mathrm{6}} \\ $$$$\Rightarrow\:\:\alpha−\beta=\frac{\pi}{\mathrm{3}} \\ $$$$\:\mathrm{sin}\:\mathrm{3}\alpha+\mathrm{sin}\:\beta \\ $$$$=\mathrm{2sin}\:\frac{\mathrm{3}\left(\alpha+\beta\right)}{\mathrm{2}}×\mathrm{cos}\:\frac{\mathrm{3}\left(\alpha−\beta\right)}{\mathrm{2}}…

Prove-I-0-x-1-pi-0-pi-e-x-cox-d-x-2-I-0-x-xI-0-x-x-2-I-0-x-0-

Question Number 219305 by Nicholas666 last updated on 22/Apr/25 $$ \\ $$$$\:\:\:\:\:\:{Prove}; \\ $$$$\:\:\:{I}_{\mathrm{0}} \left({x}\right)\:=\frac{\mathrm{1}}{\pi}\int_{\mathrm{0}} ^{\:\pi} \:{e}^{\:{x}\:{cox}\left(\theta\right)} \:{d}\theta\:; \\ $$$$\:\:\:{x}^{\mathrm{2}} {I}_{\mathrm{0}} ^{''} \left({x}\right)\:+\:{xI}'_{\mathrm{0}} \left({x}\right)\:−\:{x}^{\mathrm{2}} {I}_{\mathrm{0}}…

f-s-1-2pi-e-i-s-d-0-1-2pi-e-st-e-i-s-d-ds-

Question Number 219301 by SdC355 last updated on 22/Apr/25 $${f}\left({s}\right)=\frac{\mathrm{1}}{\mathrm{2}\pi}\:\int_{−\infty} ^{+\infty} \:\:{e}^{−\boldsymbol{{i}}\omega\left({s}−\alpha\right)} \mathrm{d}\omega\: \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{1}}{\mathrm{2}\pi}\left[{e}^{−{st}} \:\int_{\:−\infty} ^{\:+\infty} \:\:{e}^{−\boldsymbol{{i}}\omega\left({s}−\alpha\right)} \mathrm{d}\omega\right]\mathrm{d}{s}=….? \\ $$ Terms of…

E-lectric-field-strenth-at-any-point-in-the-space-is-defined-as-the-force-per-unit-charge-at-that-point-It-is-a-vector-quantity-whose-magnitude-is-given-by-Coulomb-s-law-and-diection-is-in-st

Question Number 219262 by amresh last updated on 21/Apr/25 $$\overset{} {\mathrm{E}lectric}\:\mathrm{field}\:\mathrm{strenth}\:\mathrm{at}\:\mathrm{any}\:\mathrm{point}\:\mathrm{in}\:\mathrm{the}\:\mathrm{space} \\ $$$$\mathrm{is}\:\mathrm{defined}\:\mathrm{as}\:\mathrm{the}\:\mathrm{force}\:\mathrm{per}\:\mathrm{unit}\:\mathrm{charge}\:\mathrm{at}\:\mathrm{that}\:\mathrm{point}. \\ $$$$\:\mathrm{It}\:\mathrm{is}\:\mathrm{a}\:\mathrm{vector}\:\mathrm{quantity}\:\mathrm{whose}\:\mathrm{magnitude}\:\mathrm{is} \\ $$$$\mathrm{given}\:\mathrm{by}\:\mathrm{Coulomb}^{\mathrm{s}\:\:} \:\mathrm{law}\:\mathrm{and}\:\mathrm{diection}\:\mathrm{is}\:\mathrm{in}\: \\ $$$$\mathrm{straight}\:\mathrm{line}\:\mathrm{loining}\:\mathrm{the}\:\mathrm{at}\:\mathrm{that}\:\mathrm{point}. \\ $$$$\mathrm{mathemstically} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\…

Question-219255

Question Number 219255 by Spillover last updated on 21/Apr/25 Answered by A5T last updated on 21/Apr/25 $$\mathrm{WLOG},\:\mathrm{let}\:\mathrm{the}\:\mathrm{side},\mathrm{s},\:\mathrm{of}\:\mathrm{the}\:\mathrm{square}\:\mathrm{be}\:\mathrm{1} \\ $$$$\Rightarrow\mathrm{b}^{\mathrm{2}} =\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} +\mathrm{1}^{\mathrm{2}} =\frac{\mathrm{5}}{\mathrm{4}} \\ $$$$\mathrm{a}×\mathrm{b}=\frac{\mathrm{1}}{\mathrm{2}}×\frac{\mathrm{1}}{\mathrm{2}}=\frac{\mathrm{1}}{\mathrm{4}}\overset{/\mathrm{b}^{\mathrm{2}} }…