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Category: Trigonometry

if-A-B-C-pi-so-proof-sin-A-sin-B-sin-C-4cos-A-2-cos-B-2-cos-C-2-

Question Number 21268 by oyshi last updated on 18/Sep/17 $${if}\:{A}+{B}+{C}=\pi \\ $$$${so}\:{proof}\: \\ $$$$\mathrm{sin}\:{A}+\mathrm{sin}\:{B}+\mathrm{sin}\:{C}=\mathrm{4cos}\:\frac{{A}}{\mathrm{2}}\mathrm{cos}\:\frac{{B}}{\mathrm{2}}\mathrm{cos}\:\frac{{C}}{\mathrm{2}} \\ $$ Answered by myintkhaing last updated on 18/Sep/17 $$\mathrm{L}.\mathrm{H}.\mathrm{S}=\:\mathrm{sin}\:\mathrm{A}+\mathrm{sin}\:\mathrm{B}+\mathrm{sin}\:\mathrm{C} \\…

if-subtle-and-sin-sin-3-cos-cos-so-proof-sin-3-sin-3-0-

Question Number 21260 by oyshi last updated on 17/Sep/17 $${if}\:\left(\theta−\varphi\right)\:{subtle}\:{and}\:\mathrm{sin}\:\theta+\mathrm{sin}\:\varphi=\sqrt{\mathrm{3}\left(\mathrm{cos}\:\varphi\right.} \\ $$$$\left.−\mathrm{cos}\:\theta\right) \\ $$$${so}\:{proof}\:\mathrm{sin}\:\mathrm{3}\theta+\mathrm{sin}\:\mathrm{3}\varphi=\mathrm{0} \\ $$ Answered by 951172235v last updated on 07/Feb/19 $$\mathrm{sin}\:\theta+\mathrm{sin}\:\varphi\:=\sqrt{\mathrm{3}}\:\left(\mathrm{cos}\:\varphi−\mathrm{cos}\:\theta\right) \\…

2cos-pi-13-cos-9pi-13-cos-3pi-13-cos-5pi-13-0-

Question Number 21259 by oyshi last updated on 17/Sep/17 $$\mathrm{2cos}\:\frac{\pi}{\mathrm{13}}\mathrm{cos}\:\frac{\mathrm{9}\pi}{\mathrm{13}}+\mathrm{cos}\:\frac{\mathrm{3}\pi}{\mathrm{13}}+\mathrm{cos}\:\frac{\mathrm{5}\pi}{\mathrm{13}}=\mathrm{0} \\ $$ Answered by myintkhaing last updated on 18/Sep/17 $$\mathrm{L}.\mathrm{H}.\mathrm{S}=\mathrm{cos}\:\frac{\mathrm{10}\pi}{\mathrm{13}}\:+\mathrm{cos}\:\frac{\mathrm{8}\pi}{\mathrm{13}}\:+\mathrm{cos}\:\frac{\mathrm{3}\pi}{\mathrm{13}}\:+\mathrm{cos}\:\frac{\mathrm{5}\pi}{\mathrm{13}} \\ $$$$=\mathrm{cos}\left(\pi−\frac{\mathrm{3}\pi}{\mathrm{13}}\right)+\mathrm{cos}\left(\pi−\frac{\mathrm{5}\pi}{\mathrm{13}}\right)+\mathrm{cos}\:\frac{\mathrm{3}\pi}{\mathrm{13}}+\mathrm{cos}\:\frac{\mathrm{5}\pi}{\mathrm{13}} \\ $$$$=\:−\mathrm{cos}\frac{\mathrm{3}\pi}{\mathrm{13}}−\mathrm{cos}\frac{\mathrm{5}\pi}{\mathrm{13}}+\mathrm{cos}\frac{\mathrm{3}\pi}{\mathrm{13}}+\mathrm{cos}\frac{\mathrm{5}\pi}{\mathrm{13}}\:=\:\mathrm{0} \\…

Find-y-in-3rd-quadrant-tan-y-30-cot-y-

Question Number 21241 by tawa tawa last updated on 17/Sep/17 $$\mathrm{Find}\:\mathrm{y}\:\mathrm{in}\:\mathrm{3rd}\:\mathrm{quadrant} \\ $$$$\mathrm{tan}\left(\mathrm{y}\:−\:\mathrm{30}\right)\:=\:\mathrm{cot}\left(\mathrm{y}\right) \\ $$ Answered by sma3l2996 last updated on 17/Sep/17 $${tan}\left({y}−\mathrm{30}\right)={cot}\left({y}\right) \\ $$$${tan}\left({y}−\mathrm{30}\right)=\frac{\mathrm{1}}{{tan}\left({y}\right)}\Leftrightarrow\frac{{tan}\left({y}\right)−{tan}\left(\mathrm{30}\right)}{\mathrm{1}+{tan}\left({y}\right){tan}\left(\mathrm{30}\right)}=\frac{\mathrm{1}}{{tany}}…

cos-pi-17-cos-2pi-17-cos-4pi-17-cos-8pi-17-

Question Number 86716 by john santu last updated on 30/Mar/20 $$\mathrm{cos}\:\left(\frac{\pi}{\mathrm{17}}\right)×\mathrm{cos}\:\left(\frac{\mathrm{2}\pi}{\mathrm{17}}\right)×\mathrm{cos}\:\left(\frac{\mathrm{4}\pi}{\mathrm{17}}\right)×\mathrm{cos}\:\left(\frac{\mathrm{8}\pi}{\mathrm{17}}\right)\:= \\ $$ Commented by jagoll last updated on 30/Mar/20 $$\frac{\mathrm{cos}\:\mathrm{x}\:\mathrm{cos}\:\mathrm{2x}\:\mathrm{cos}\:\mathrm{4x}\:\mathrm{cos}\:\mathrm{8x}}{\mathrm{2sin}\:\mathrm{x}}\:×\:\mathrm{2sin}\:\mathrm{x}\:\: \\ $$$$\frac{\mathrm{sin}\:\mathrm{2x}\:\mathrm{cos}\:\mathrm{2x}\:\mathrm{cos}\:\mathrm{4x}\:\mathrm{cos}\:\mathrm{8x}}{\mathrm{2sin}\:\mathrm{x}}\:=\: \\ $$$$\frac{\mathrm{sin}\:\mathrm{4x}\:\mathrm{cos}\:\mathrm{4x}\:\mathrm{cos}\:\mathrm{8x}}{\mathrm{4sin}\:\mathrm{x}}\:=\:…

find-solution-4-sin-x-1-4-1-2-2-2-sin-x-1-0-in-x-0-2pi-

Question Number 86701 by john santu last updated on 30/Mar/20 $$\mathrm{find}\:\mathrm{solution}\: \\ $$$$\mathrm{4}^{\mathrm{sin}\:\mathrm{x}\:−\frac{\mathrm{1}}{\mathrm{4}}} \:−\:\frac{\mathrm{1}}{\mathrm{2}+\sqrt{\mathrm{2}}}\:.\mathrm{2}^{\mathrm{sin}\:\mathrm{x}} \:−\mathrm{1}\:=\:\mathrm{0}\: \\ $$$$\mathrm{in}\:\mathrm{x}\:\in\left[\:\mathrm{0},\mathrm{2}\pi\:\right]\: \\ $$ Commented by john santu last updated…

if-sin-sin-cos-cos-1-0-so-proof-that-1-cot-tan-0-

Question Number 21141 by oyshi last updated on 14/Sep/17 $${if}\:\mathrm{sin}\:\alpha\mathrm{sin}\:\beta−\mathrm{cos}\:\alpha\mathrm{cos}\:\beta+\mathrm{1}=\mathrm{0}\: \\ $$$${so}\:{proof}\:{that}\:\mathrm{1}+\mathrm{cot}\:\alpha\mathrm{tan}\:\beta=\mathrm{0} \\ $$ Answered by $@ty@m last updated on 15/Sep/17 $$\mathrm{sin}\:\alpha\mathrm{sin}\:\beta−\mathrm{cos}\:\alpha\mathrm{cos}\:\beta+\mathrm{1}=\mathrm{0}\: \\ $$$$\mathrm{cos}\:\alpha\mathrm{cos}\:\beta−\mathrm{sin}\:\alpha\mathrm{sin}\:\beta=\mathrm{1} \\…