Category: Trigonometry

Question Number 184932 by mathlove last updated on 14/Jan/23 Answered by qaz last updated on 14/Jan/23 $$\underset{n=1}{\sum ^{\mathrm{\infty }}}\frac{1}{n}\sin \frac{n\pi }{3}=\mathrm{\Im }\underset{n=1}{\sum ^{\mathrm{\infty }}}\frac{e^{i\frac{n\pi }{3}}}{n}=-\mathrm{\Im }ln(1-e^{\frac{i\pi }{3}})$$$$=−\Im{ln}\left(\frac{\mathrm{1}}{\mathrm{2}}−\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}{i}\right)=−\Im{lne}^{−{i}\mathrm{arctan}\:\sqrt{\mathrm{3}}}…

Question Number 119372 by bobhans last updated on 24/Oct/20 $$Determineminimumvalueof$$$$\frac{\sec {}^4\alpha }{\tan {}^2\beta }+\frac{\sec {}^4\beta }{\tan {}^2\alpha },overall\alpha ,\beta \ne \frac{k\pi }{2}$$$$andk\in \mathbb{Z}$$ Answered by 1549442205PVT last updated…

Question Number 119266 by benjo_mathlover last updated on 23/Oct/20 $$solve\frac{\sqrt{3}-1}{\sin x}+\frac{\sqrt{3}+1}{\cos x}=4\sqrt{2}$$$$wherex\in (0,\frac{\pi }{2})$$ Commented by MJS_new last updated on 23/Oct/20 $$(1+\sqrt{3})\sin x-(1-\sqrt{3})\cos x=4\sqrt{2}\sin x\cos x$$$$\:\:\:\:\:\left[\mathrm{sorry}\:\mathrm{no}\:\mathrm{time}\:\mathrm{to}\:\mathrm{type}\:\mathrm{this}\:\mathrm{transformation}\right] \…

Question Number 119229 by benjo_mathlover last updated on 23/Oct/20 $$\sin 3A+\cos 3A=\frac{1}{2}$$ Answered by bemath last updated on 23/Oct/20 $$let3A=x;\cos x=Xand\sin x=Y$$$$\Rightarrow{X}+{Y}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\:;\:{X}^{\mathrm{2}} +\mathrm{2}{XY}\:+{Y}^{\mathrm{2}} \:=\:\frac{\mathrm{1}}{\mathrm{4}} \…

Question Number 119109 by bagjagunawan last updated on 22/Oct/20 Answered by 1549442205PVT last updated on 22/Oct/20 $$\mathrm{P}=(\frac{1}{2}+\cos \frac{\pi }{20})(\frac{1}{2}+\cos \frac{3\pi }{20})(\frac{1}{2}+\cos \frac{9\pi }{20})(\frac{1}{2}+\cos \frac{27\pi }{20})$$$$=(\frac{1}{2}+\cos 9)(\frac{1}{2}+\cos 81)(\frac{1}{2}+\cos 27)(\frac{1}{2}-\cos 63)$$$$=(\frac{1}{4}+\frac{\cos 9+\cos 81}{2}+\cos 9\cos 81)$$$$\times (\frac{1}{4}+\frac{\cos 27-\cos 63}{2}-\cos 27\cos 63)$$$$=\left(\frac{\mathrm{1}}{\mathrm{4}}+\mathrm{cos45cos36}+\frac{\mathrm{cos90}+\mathrm{cos72}}{\mathrm{2}}\right)…