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Category: Trigonometry

Question-7514

Question Number 7514 by Tawakalitu. last updated on 01/Sep/16 Answered by Yozzia last updated on 01/Sep/16 $${sin}\left(\pi{cos}\alpha\right)={cos}\left(\pi{sin}\alpha\right). \\ $$$${Using}\:{sina}={cos}\left(\frac{\pi}{\mathrm{2}}−{a}\right),\:{we}\:{get} \\ $$$${cos}\left(\frac{\pi}{\mathrm{2}}−\pi{cos}\alpha\right)={cos}\left(\pi{sin}\alpha\right) \\ $$$$\Rightarrow\frac{\pi}{\mathrm{2}}−\pi{cos}\alpha=\mathrm{2}{n}\pi\pm\pi{sin}\alpha\:\:\:\:\left({n}\in\mathbb{Z}\right) \\ $$$$\mp\pi{sin}\alpha−\pi{cos}\alpha=\mathrm{2}{n}\pi−\frac{\pi}{\mathrm{2}}…

x-5-x-5-tan-find-the-value-of-

Question Number 7392 by Tawakalitu. last updated on 26/Aug/16 $$\sqrt{{x}}\:+\:\mathrm{5}\:+\:\sqrt{{x}}\:+\:\mathrm{5}\:=\:{tan}\Theta \\ $$$$ \\ $$$${find}\:{the}\:{value}\:{of}\:\Theta \\ $$ Commented by Yozzia last updated on 26/Aug/16 $${general}\:{solution},\:\Theta={n}\pi+{tan}^{−\mathrm{1}} \left(\mathrm{2}\sqrt{{x}}+\mathrm{10}\right),\:{n}\in\mathbb{Z}.…

The-sides-of-a-triangle-are-x-cm-x-4-cm-x-8-cm-respectively-if-the-cosine-o-the-largest-is-1-5-calculate-the-angles-of-triangle-

Question Number 7142 by Tawakalitu. last updated on 13/Aug/16 $${The}\:{sides}\:{of}\:{a}\:{triangle}\:{are}\:{x}\:{cm},\:\left({x}\:−\:\mathrm{4}\right)\:{cm},\:\left({x}\:−\:\mathrm{8}\right){cm} \\ $$$${respectively}\:.\:{if}\:{the}\:{cosine}\:{o}\:{the}\:{largest}\:{is}\:\frac{\mathrm{1}}{\mathrm{5}}\:,\: \\ $$$${calculate}\:{the}\:{angles}\:{of}\:\:{triangle}. \\ $$$$ \\ $$ Answered by Rasheed Soomro last updated on…

Show-that-x-0-2-tanx-sin4x-0-

Question Number 7055 by Tawakalitu. last updated on 08/Aug/16 $${Show}\:{that}\:\forall\:{x}\:\in\:\left[\:\mathrm{0},\:\frac{\Pi}{\mathrm{2}}\:\right]\:,\:{tanx}\:+\:{sin}\mathrm{4}{x}\:\geqslant\:\mathrm{0}\: \\ $$$$ \\ $$ Commented by Yozzii last updated on 08/Aug/16 $${If}\:\mathrm{0}\leqslant{x}\leqslant\frac{\pi}{\mathrm{2}}\Rightarrow\mathrm{0}\leqslant\mathrm{4}{x}\leqslant\mathrm{2}\pi \\ $$$${For}\:−\mathrm{1}\leqslant{sin}\mathrm{4}{x}<\mathrm{0}\Rightarrow\pi<\mathrm{4}{x}\leqslant\frac{\mathrm{3}\pi}{\mathrm{2}}\:{or}\:\frac{\mathrm{3}\pi}{\mathrm{2}}\leqslant\mathrm{4}{x}<\mathrm{2}\pi\: \\…

Show-without-using-calculator-that-tan-1-sin-20sin-150sin-160-sin-10-sin-140-sin-20-sin-150-cos-160-130-

Question Number 7039 by Rasheed Soomro last updated on 07/Aug/16 $${Show}\:{without}\:{using}\:{calculator}\:{that} \\ $$$$\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{sin}\:\mathrm{20sin}\:\mathrm{150sin}\:\mathrm{160}}{\mathrm{sin}\:\mathrm{10}\:\mathrm{sin}\:\mathrm{140}+\mathrm{sin}\:\mathrm{20}\:\mathrm{sin}\:\mathrm{150}\:\mathrm{cos}\:\mathrm{160}}\right)=\mathrm{130} \\ $$ Commented by Yozzii last updated on 08/Aug/16 $${All}\:{figures}\:{are}\:{assumed}\:{to}\:{be}\:{in}\:{degrees}. \\…

If-sin-4-cos-4-4sin-cos-0-pi-2-then-sin-cos-equal-to-

Question Number 138099 by liberty last updated on 10/Apr/21 $${If}\:\mathrm{sin}\:^{\mathrm{4}} \alpha+\mathrm{cos}\:^{\mathrm{4}} \beta\:=\:\mathrm{4sin}\:\alpha\:\mathrm{cos}\:\beta\:, \\ $$$$\mathrm{0}\leqslant\alpha,\beta\:\leqslant\:\frac{\pi}{\mathrm{2}}\:,\:{then}\:\mathrm{sin}\:\alpha+\mathrm{cos}\:\beta\: \\ $$$${equal}\:{to}\:\ldots \\ $$ Commented by mr W last updated on…

if-cot-cosec-1-3-then-find-the-value-of-where-0-lt-2pi-

Question Number 72497 by Shamim last updated on 29/Oct/19 $$\mathrm{if},\:\mathrm{cot}\:\theta+\mathrm{cosec}\:\theta=\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}\:\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value} \\ $$$$\mathrm{of}\:\theta\:\mathrm{where}\:\mathrm{0}<\theta\leqslant\mathrm{2}\pi. \\ $$ Answered by behi83417@gmail.com last updated on 29/Oct/19 $$\frac{\mathrm{cos}\theta}{\mathrm{sin}\theta}+\frac{\mathrm{1}}{\mathrm{sin}\theta}=\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}\Rightarrow\frac{\mathrm{1}+\mathrm{cos}\theta}{\mathrm{sin}\theta}=\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}\Rightarrow \\ $$$$\frac{\mathrm{2cos}^{\mathrm{2}} \frac{\theta}{\mathrm{2}}}{\mathrm{2sin}\frac{\theta}{\mathrm{2}}\mathrm{cos}\frac{\theta}{\mathrm{2}}}=\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}\Rightarrow\begin{cases}{\mathrm{1}.\mathrm{cos}\frac{\theta}{\mathrm{2}}=\mathrm{0}}\\{\mathrm{2}.\mathrm{cot}\frac{\theta}{\mathrm{2}}=\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}}\end{cases}…