Question and Answers Forum

All Questions      Topic List

Trigonometry Questions

Previous in All Question      Next in All Question      

Previous in Trigonometry      Next in Trigonometry      

Question Number 18962 by Tinkutara last updated on 02/Aug/17

$$\mathrm{The}\mathrm{number}\mathrm{of}\mathrm{solutions}\mathrm{of}\mathrm{the}\mathrm{equation}$$$${\sin }^5\theta +\frac{1}{\sin \theta }=\frac{1}{\cos \theta }+{\cos }^5\theta \mathrm{where}$$$$\theta \in (0,\frac{\pi }{2}),\mathrm{is}$$

Answered by behi.8.3.4.1.7@gmail.com last updated on 02/Aug/17

$${sin}^6\theta .cos\theta +cos\theta =sin\theta +{cos}^6\theta .sin\theta$$$$\Rightarrow sin\theta .cos\theta ({sin}^5\theta -{cos}^5\theta )-(sin\theta -cos\theta )=0$$$$sin\theta =a,cos\theta =b$$$$\Rightarrow ab(a^5-b^5)-(a-b)=0$$$$\Rightarrow [ab(a^4+a^{3}b+a^2{b}^2+{ab}^3+b^4)-1](a-b)=0$$$$1)a=b\Rightarrow sin\theta =cos\theta \Rightarrow tg\theta =1\Rightarrow \theta =k\pi +\frac{\pi }{4},k\in Z$$$$2)a^4+b^4={(a^2+b^2)}^2-2a^2{b}^2=1-2a^2{b}^2$$$$a^{3}b+{ab}^3=ab(a^2+b^2)=ab$$$$\Rightarrow ab(1-2a^2{b}^2+a^2{b}^2+ab)-1=0$$$$\Rightarrow ab(1+ab-a^2{b}^2)-1=0(ab=t)$$$$\Rightarrow t(1+t-t^2)-1=0\Rightarrow -t^3+t^2+t-1=0$$$$\Rightarrow t^3-t^2-t+1=0\Rightarrow t^2(t-1)-(t-1)=0$$$$\Rightarrow (t-1)(t^2-1)=0\Rightarrow {(t-1)}^2(t+1)=0$$$$\Rightarrow t=1,-1\Rightarrow sin\theta .cos\theta =\pm 1\Rightarrow sin2\theta =\pm 2$$$$thiseq.hasnotrealanswers.$$

Commented by Tinkutara last updated on 02/Aug/17

$$\mathrm{Thank}\mathrm{you}\mathrm{very}\mathrm{much}\mathrm{behi}\mathrm{Sir}!$$

Commented by behi.8.3.4.1.7@gmail.com last updated on 04/Aug/17

$$sin2\theta =2\Rightarrow \frac{e^{2\theta }-e^{-2\theta }}{2i}=2(e^{2\theta }=a)$$$$\Rightarrow \frac{a-a^{-1}}{2i}=2\Rightarrow a^2-4ai-1=0$$$$\Rightarrow a=\frac{4i\pm \sqrt{-16+4}}{2}=\frac{4i\pm 2\sqrt{3}i}{2}=(2\pm \sqrt{3})i$$$$\Rightarrow e^{2\theta }=(2\pm \sqrt{3})i\Rightarrow 2\theta =ln(2\pm \sqrt{3})+lni$$$$\Rightarrow \theta =\frac{1}{2}ln(2\pm \sqrt{3})+\frac{i\pi }{4}\left(complexanswer\right).$$

Commented by Tinkutara last updated on 04/Aug/17

$$\mathrm{But}\mathrm{using}\mathrm{this}\mathrm{value}\mathrm{in}\mathrm{calculator}\mathrm{does}$$$$\mathrm{not}\mathrm{gives}2.\sin (\ln (2+\sqrt{3})+\frac{i\pi }{2})$$$$=2.423+0.578i$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com