Determine-the-value-of-S-pi-if-S-is-the-sum-in-radians-all-equation-solutions-contained-in-the-interval-0-14pi-the-equation-is-cos-x-cos-5-x-cos-7x-3-

FacebookTweetPin

Question Number 131097 by dw last updated on 01/Feb/21

$$Determinethevalueof\frac{S}{\pi },ifSisthesum,inradians,$$$$allequationsolutionscontainedintheinterval[0,14\pi ].$$$$$$$$theequationis:$$$$cos\left(x\right)+{cos}^5\left(x\right)+cos\left(7x\right)=3$$

Answered by MJS_new last updated on 01/Feb/21

$$\mathrm{the}\mathrm{only}\mathrm{solution}\mathrm{is}\cos x=1$$$$\Rightarrow x=2n\pi$$$$\Rightarrow S=2\pi (0+1+2+\dots +14)=210\pi$$$$\Rightarrow \mathrm{answer}\mathrm{is}210$$

Commented by dw last updated on 01/Feb/21

$$Si\overline{r},howcouldIproveit?$$

Commented by dw last updated on 01/Feb/21

$$Thankyouforyoursolution.$$

Answered by MJS_new last updated on 01/Feb/21

$$\cos x+{\cos }^{5}x+\cos 7x-3=0$$$$\mathrm{with}$$$$\cos 7x={64\cos }^{7}x-{112\cos }^{5}x+{56\cos }^{3}x-7\cos x$$$$\mathrm{and}$$$$c=\cos x$$$$\mathrm{we}\mathrm{have}$$$$64c^7-111c^5+56c^3-6c-3=0$$$$\mathrm{trying}\mathrm{factors}\mathrm{of}\mathrm{the}\mathrm{constant}\{\pm 1,\pm 3\}\mathrm{we}$$$$\mathrm{get}$$$$c_1=1$$$$f=64c^6+64c^5-47c^4-47c^3+9c^2+9c+3$$$$\mathrm{this}\mathrm{has}\mathrm{no}\mathrm{real}\mathrm{solution}$$$$f^{\prime }=384c^5+320c^4-188c^3-141c^2+18c+9$$$$\mathrm{this}\mathrm{has}\mathrm{the}\mathrm{zeros}$$$$c_1\approx -.891\Rightarrow f\approx 1.83$$$$c_2\approx -.608\Rightarrow f\approx 2.91$$$$c_3\approx -.238\Rightarrow f\approx 1.81$$$$c_4\approx .302\Rightarrow f\approx 5.06$$$$c_5\approx .601\Rightarrow f\approx 3.36$$$$\Rightarrow \mathrm{the}\mathrm{absolute}\mathrm{minimum}\mathrm{of}f\mathrm{is}\approx 1.81$$$$\Rightarrow \mathrm{no}\mathrm{more}\mathrm{zeros}$$$$\Rightarrow$$$$c=1$$

Commented by dw last updated on 01/Feb/21

$$ThankyouSir!!$$

Terms of Service
Privacy Policy
Contact: info@tinkutara.com

FacebookTweetPin