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Category: Trigonometry

CosA-CosB-CosC-1-4Cos-B-C-2-Cos-C-A-2-Cos-A-B-2-1-4Cos-A-4-Cos-B-4-Cos-C-4-prove-that-if-A-B-C-

Question Number 67083 by lalitchand last updated on 22/Aug/19 $$\mathrm{CosA}+\mathrm{CosB}+\mathrm{CosC}=\mathrm{1}+\mathrm{4Cos}\left(\frac{\mathrm{B}+\mathrm{C}}{\mathrm{2}}\right).\mathrm{Cos}\left(\frac{\mathrm{C}+\mathrm{A}}{\mathrm{2}}\right).\mathrm{Cos}\left(\frac{\mathrm{A}+\mathrm{B}}{\mathrm{2}}\right)=\mathrm{1}+\mathrm{4Cos}\left(\frac{\Pi−\mathrm{A}}{\mathrm{4}}\right).\mathrm{Cos}\left(\frac{\Pi−\mathrm{B}}{\mathrm{4}}\right).\mathrm{Cos}\left(\frac{\Pi−\mathrm{C}}{\mathrm{4}}\right) \\ $$$$\mathrm{prove}\:\mathrm{that}\:\mathrm{if}\:\mathrm{A}+\mathrm{B}+\mathrm{C}=\Pi \\ $$ Answered by Tanmay chaudhury last updated on 23/Aug/19 $${LHS} \\ $$$$\mathrm{2}{cos}\left(\frac{{A}+{B}}{\mathrm{2}}\right){cos}\left(\frac{{A}−{B}}{\mathrm{2}}\right)+\mathrm{1}−\mathrm{2}{sin}^{\mathrm{2}}…

Find-minimum-and-maximum-value-of-function-f-x-2sin-x-3-sin-x-1-

Question Number 132531 by bemath last updated on 15/Feb/21 $$\mathrm{Find}\:\mathrm{minimum}\:\mathrm{and}\:\mathrm{maximum} \\ $$$$\mathrm{value}\:\mathrm{of}\:\mathrm{function}\:\mathrm{f}\left(\mathrm{x}\right)=\sqrt{\mathrm{2sin}\:\mathrm{x}+\mathrm{3}}−\sqrt{\mathrm{sin}\:\mathrm{x}+\mathrm{1}} \\ $$ Answered by liberty last updated on 15/Feb/21 $$\frac{\mathrm{df}\left(\mathrm{x}\right)}{\mathrm{dx}}=\frac{\mathrm{cos}\:\mathrm{x}}{\:\sqrt{\mathrm{2sin}\:\mathrm{x}+\mathrm{3}}}−\frac{\mathrm{cos}\:\mathrm{x}}{\mathrm{2}\sqrt{\mathrm{sin}\:\mathrm{x}+\mathrm{1}}}\:=\mathrm{0} \\ $$$$\:\frac{\mathrm{cos}\:\mathrm{x}}{\:\sqrt{\mathrm{2sin}\:\mathrm{x}+\mathrm{3}}}\:=\:\frac{\mathrm{cos}\:\mathrm{x}}{\mathrm{2}\sqrt{\mathrm{sin}\:\mathrm{x}+\mathrm{1}}} \\…

Solve-the-following-inequality-sinx-1-cosx-1-where-0-x-lt-2pi-cosx-0-

Question Number 1407 by 112358 last updated on 29/Jul/15 $${Solve}\:{the}\:{following}\:{inequality} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{{sinx}+\mathrm{1}}{{cosx}}\leqslant\mathrm{1} \\ $$$${where}\:\mathrm{0}\leqslant{x}<\mathrm{2}\pi\:,\:{cosx}\neq\mathrm{0} \\ $$ Commented by 123456 last updated on 29/Jul/15 $${f}\left({x}\right)=\frac{\mathrm{sin}\:{x}+\mathrm{1}}{\mathrm{cos}\:{x}} \\…

sin-2x-pi-4-sin-2x-pi-3-0-x-

Question Number 132347 by Study last updated on 13/Feb/21 $${sin}\left(\mathrm{2}{x}+\frac{\pi}{\mathrm{4}}\right)−{sin}\left(\mathrm{2}{x}+\frac{\pi}{\mathrm{3}}\right)=\mathrm{0}\:\:\:{x}=? \\ $$ Answered by EDWIN88 last updated on 13/Feb/21 $$\mathrm{sin}\:\mathrm{X}−\mathrm{sin}\:\mathrm{Y}\:=\:\mathrm{2cos}\:\left(\frac{\mathrm{X}+\mathrm{Y}}{\mathrm{2}}\right)\mathrm{sin}\:\:\left(\frac{\mathrm{X}−\mathrm{Y}}{\mathrm{2}}\right) \\ $$$$\mathrm{where}\:\begin{cases}{\mathrm{X}=\mathrm{2x}+\frac{\pi}{\mathrm{4}}}\\{\mathrm{Y}=\mathrm{2x}+\frac{\pi}{\mathrm{3}}}\end{cases} \\ $$$$\mathrm{sin}\:\left(\mathrm{2x}+\frac{\pi}{\mathrm{4}}\right)−\mathrm{sin}\:\left(\mathrm{2x}+\frac{\pi}{\mathrm{3}}\right)=\mathrm{0} \\…

without-using-mathematical-tables-evaluate-sin-60-tan-30-cos-60-sin-30-cos-45-sin-45-sin-90-cos-45-sin-45-sin-60-cos-30-sin-30-

Question Number 66768 by John Kaloki Musau last updated on 19/Aug/19 $${without}\:{using}\:{mathematical}\:{tables} \\ $$$${evaluate} \\ $$$$\frac{\mathrm{sin}\:\mathrm{60}.\mathrm{tan}\:\mathrm{30}.\mathrm{cos}\:\mathrm{60}+\mathrm{sin}\:\mathrm{30}.\mathrm{cos}\:\mathrm{45}.\mathrm{sin}\:\mathrm{45}}{\mathrm{sin}\:\mathrm{90}.\mathrm{cos}\:\mathrm{45}.\mathrm{sin}\:\mathrm{45}−\mathrm{sin}\:\mathrm{60}.\mathrm{cos}\:\mathrm{30}.\mathrm{sin}\:\mathrm{30}} \\ $$ Commented by Tony Lin last updated on…