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Category: Geometry

If-p-cos-i-sin-and-q-cos-i-sin-Show-that-i-p-q-p-q-i-tan-2-ii-p-q-pq-1-p-q-pq-1-sin-sin-sin

Question Number 51905 by Tawa1 last updated on 01/Jan/19 $$\mathrm{If}\:\:\:\mathrm{p}\:\:=\:\:\mathrm{cos}\:\theta\:+\:\mathrm{i}\:\mathrm{sin}\theta\:\:\:\:\:\:\:\:\:\:\:\mathrm{and}\:\:\:\:\:\:\:\mathrm{q}\:\:=\:\:\mathrm{cos}\:\phi\:+\:\mathrm{i}\:\mathrm{sin}\:\phi \\ $$$$\mathrm{Show}\:\mathrm{that}: \\ $$$$\left(\mathrm{i}\right)\:\:\:\:\:\:\frac{\mathrm{p}\:−\:\mathrm{q}}{\mathrm{p}\:+\:\mathrm{q}}\:\:=\:\:\mathrm{i}\:\mathrm{tan}\:\left(\frac{\theta\:−\:\phi}{\mathrm{2}}\right) \\ $$$$\left(\mathrm{ii}\right)\:\:\:\frac{\left(\mathrm{p}\:+\:\mathrm{q}\right)\left(\mathrm{pq}\:−\:\mathrm{1}\right)}{\left(\mathrm{p}\:−\:\mathrm{q}\right)\left(\mathrm{pq}\:+\:\mathrm{1}\right)}\:\:=\:\:\frac{\mathrm{sin}\:\theta\:+\:\mathrm{sin}\:\phi}{\mathrm{sin}\:\theta\:−\:\mathrm{sin}\:\phi} \\ $$ Answered by Kunal12588 last updated on 01/Jan/19…

Question-51884

Question Number 51884 by ajfour last updated on 31/Dec/18 Commented by ajfour last updated on 31/Dec/18 $${Relate}\:{radius}\:\boldsymbol{{r}}\:{of}\:{the}\:{two}\:{circles} \\ $$$$\left({the}\:{same}\right)\:{with}\:{side}\:{length}\:\boldsymbol{{a}}\:{of} \\ $$$${the}\:{equilateral}\:{triangle}. \\ $$ Commented by…

Question-182910

Question Number 182910 by mnjuly1970 last updated on 16/Dec/22 Answered by mr W last updated on 16/Dec/22 $${a}={side}\:{length}\:{of}\:{square} \\ $$$${R}={radius}\:{of}\:{circle} \\ $$$${R}^{\mathrm{2}} =\left({R}−{a}\right)^{\mathrm{2}} +\left({R}−\frac{{a}}{\mathrm{2}}\right)^{\mathrm{2}} \\…

Question-51700

Question Number 51700 by mr W last updated on 29/Dec/18 Commented by mr W last updated on 29/Dec/18 $${A}\:{rope}\:{with}\:{uniform}\:{mass}\:{is}\:{fixed} \\ $$$${at}\:{one}\:{end}\:{on}\:{point}\:{A}.\:{At}\:{point}\:{B}\:{the} \\ $$$${rope}\:{is}\:{laid}\:{over}\:{a}\:{pulley}.\:{All}\:{contact} \\ $$$${is}\:{frictionless}\:{and}\:{the}\:{size}\:{of}\:{pulley}…