Question Number 70394 by Cmr 237 last updated on 04/Oct/19 $$\mathrm{montrer}\:\mathrm{que} \\ $$$$\mathrm{sinA}+\mathrm{sinB}+\mathrm{sinC}=\mathrm{4cos}\frac{\mathrm{A}}{\mathrm{2}}\mathrm{cos}\frac{\mathrm{B}}{\mathrm{2}}\mathrm{cos}\frac{\mathrm{C}}{\mathrm{2}} \\ $$ Answered by $@ty@m123 last updated on 05/Oct/19 $${LHS}=\mathrm{2sin}\:\frac{{A}+{B}}{\mathrm{2}}\mathrm{cos}\:\frac{{A}−{B}}{\mathrm{2}}+\mathrm{sin}\:{C} \\ $$$$=\mathrm{2sin}\:\frac{\pi−{C}}{\mathrm{2}}\mathrm{cos}\:\frac{{A}−{B}}{\mathrm{2}}+\mathrm{sin}\:{C}…
Question Number 135926 by liberty last updated on 17/Mar/21 $$ \\ $$If cos(A+B) =1/2, sinA=1/√2, then what is cos (A – B)? Answered by benjo_mathlover…
Question Number 135895 by liberty last updated on 16/Mar/21 $${Trigonometry} \\ $$If sin A + sin B = a and cos A + cos B…
Question Number 135868 by liberty last updated on 16/Mar/21 Answered by EDWIN88 last updated on 16/Mar/21 $$\mathrm{we}\:\mathrm{know}\:\mathrm{that}\:−\mathrm{1}\leqslant\mathrm{cos}\:\mathrm{x}\leqslant\mathrm{1}\:\mathrm{for}\:\mathrm{x}\:\mathrm{in}\:\mathrm{the}\:\mathrm{interval} \\ $$$$\mathrm{0}\leqslant\mathrm{x}\leqslant\mathrm{2}\pi\:;\:\mathrm{so}\:\mathrm{we}\:\mathrm{find}\:\mathrm{cos}\:\mathrm{x}\:=\:\frac{−\mathrm{x}^{\mathrm{3}} −\mathrm{2x}^{\mathrm{2}} −\mathrm{5x}}{\mathrm{2}} \\ $$$$\mathrm{then}\:−\mathrm{2}\leqslant−\mathrm{x}^{\mathrm{3}} −\mathrm{2x}^{\mathrm{2}} −\mathrm{5x}\leqslant\mathrm{2}\:\mathrm{or}\:…
Question Number 135855 by liberty last updated on 16/Mar/21 $${What}\:{are}\:{the}\:{possible}\:{value}\:{of} \\ $$$$\mathrm{cos}\:\alpha×\mathrm{sin}\:\beta\:\:{if}\:\mathrm{sin}\:\alpha×\mathrm{cos}\:\beta=−\frac{\mathrm{1}}{\mathrm{2}} \\ $$ Answered by EDWIN88 last updated on 16/Mar/21 $$\mathrm{We}\:\mathrm{have}\:−\mathrm{1}\leqslant\mathrm{sin}\:\left(\alpha+\beta\right)\leqslant\mathrm{1}\:\mathrm{and}\:−\mathrm{1}\leqslant\mathrm{sin}\:\left(\alpha−\beta\right)\leqslant\mathrm{1} \\ $$$$\mathrm{now}\:\mathrm{from}\:−\mathrm{1}\leqslant\mathrm{sin}\:\left(\alpha+\beta\right)\leqslant\mathrm{1}\:\mathrm{we}\:\mathrm{get}\: \\…
Question Number 135792 by benjo_mathlover last updated on 16/Mar/21 $${Find}\:{the}\:{minimum}\:{value} \\ $$$${of}\:\mid\mathrm{sin}\:{x}+\mathrm{cos}\:{x}+\mathrm{tan}\:{x}+\mathrm{cot}\:{x}+\mathrm{sec}\:{x}+\mathrm{csc}\:{x}\mid\: \\ $$$${for}\:{real}\:{numbers}\:{x}. \\ $$ Answered by MJS_new last updated on 16/Mar/21 $${x}={t}+\frac{\mathrm{5}\pi}{\mathrm{4}}\:\mathrm{leads}\:\mathrm{to} \\…
Question Number 135795 by benjo_mathlover last updated on 16/Mar/21 $${What}\:{the}\:{value}\:{of}\:\left(\mathrm{1}−\mathrm{cot}\:\mathrm{23}°\right)\left(\mathrm{1}−\mathrm{cot}\:\mathrm{22}°\right). \\ $$ Answered by MJS_new last updated on 16/Mar/21 $$\left(\mathrm{1}−\mathrm{cot}\:\left(\frac{\mathrm{45}}{\mathrm{2}}−{x}\right)\right)\left(\mathrm{1}−\mathrm{cot}\:\left(\frac{\mathrm{45}}{\mathrm{2}}+{x}\right)\right)=\mathrm{2} \\ $$ Answered by liberty…
Question Number 135791 by benjo_mathlover last updated on 16/Mar/21 $${If}\:{x}^{\mathrm{2}} +\left(\mathrm{tan}\:\theta+\mathrm{cot}\:\theta\right){x}\:+\:\mathrm{1}\:=\:\mathrm{0} \\ $$$${has}\:{two}\:{real}\:{solutions}\: \\ $$$$\left\{\:\mathrm{2}−\sqrt{\mathrm{3}}\:,\:\mathrm{2}+\sqrt{\mathrm{3}}\:\right\},\:{find}\:\begin{cases}{\mathrm{sin}\:\theta}\\{\mathrm{cos}\:\theta}\end{cases}\:. \\ $$ Answered by EDWIN88 last updated on 17/Mar/21 $$\mathrm{By}\:\mathrm{Vieta}'\mathrm{s}\:\Rightarrow\:\mathrm{x}_{\mathrm{1}}…
Question Number 135679 by liberty last updated on 15/Mar/21 $$\left(\mathrm{cos}\:{x}−\mathrm{sin}\:{x}\right)\left(\mathrm{2tan}\:{x}+\frac{\mathrm{1}}{\mathrm{cos}\:{x}}\right)+\mathrm{2}\:=\:\mathrm{0} \\ $$ Answered by EDWIN88 last updated on 15/Mar/21 $$\mathrm{Let}\:\mathrm{tan}\:\frac{\mathrm{x}}{\mathrm{2}}\:=\:\mathrm{t}\:\rightarrow\begin{cases}{\mathrm{cos}\:\mathrm{x}=\frac{\mathrm{1}−\mathrm{t}}{\mathrm{1}+\mathrm{t}^{\mathrm{2}} }}\\{\mathrm{sin}\:\mathrm{x}=\frac{\mathrm{2t}}{\mathrm{1}+\mathrm{t}^{\mathrm{2}} }}\\{\mathrm{tan}\:\mathrm{x}=\frac{\mathrm{2t}}{\mathrm{1}−\mathrm{t}^{\mathrm{2}} }}\end{cases} \\ $$$$\Leftrightarrow\:\left(\frac{\mathrm{1}−\mathrm{t}^{\mathrm{2}}…
Question Number 4600 by Yozzii last updated on 11/Feb/16 $${Find}\:{the}\:{value}\:{of}\:{S}\:{where}\:{S}=\underset{{r}=\mathrm{1}} {\overset{\mathrm{89}} {\sum}}{sin}^{\mathrm{6}} {r}°. \\ $$ Commented by FilupSmith last updated on 11/Feb/16 $$\mathrm{Don}'\mathrm{t}\:\mathrm{you}\:\mathrm{use}\:\mathrm{the}\:\mathrm{series}\:\mathrm{representation} \\ $$$$\mathrm{for}\:\mathrm{sine}?…