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 144828 by mathlove last updated on 29/Jun/21

2sin 17x+(√3) cos 5x+sin 5x=0 x=?

$$\mathrm{2sin}\:\mathrm{17}{x}+\sqrt{\mathrm{3}}\:\mathrm{cos}\:\mathrm{5}{x}+\mathrm{sin}\:\mathrm{5}{x}=\mathrm{0} \\ $$$${x}=? \\ $$

Answered by liberty last updated on 30/Jun/21

2sin 17x + 2cos (5x−(π/6))=0 ⇒cos (5x−(π/6))=−sin 17x ⇒cos (5x−(π/6))=cos ((π/2)+17x) ⇒2kπ ±( 5x−(π/6))=(π/2)+17x ⇒2kπ∓(π/6)−(π/2)=17x ∓ 5x ⇒2kπ−((2π)/3)= 17x ∓ 5x (case 1)2kπ−((2π)/3)= 12x ⇒x = ((kπ)/6)−(π/(18)) (case 2) 2kπ−((2π)/3)= 22x ⇒x = ((kπ)/(11))−(π/(33)). (case 3) 2kπ−(π/3)=12x ⇒x=((kπ)/6)−(π/(36)) (case 4) 2kπ−(π/3)=22x ⇒x=((kπ)/(11))−(π/(66))

$$\mathrm{2sin}\:\mathrm{17x}\:+\:\mathrm{2cos}\:\left(\mathrm{5x}−\frac{\pi}{\mathrm{6}}\right)=\mathrm{0} \\ $$$$\Rightarrow\mathrm{cos}\:\left(\mathrm{5x}−\frac{\pi}{\mathrm{6}}\right)=−\mathrm{sin}\:\mathrm{17x} \\ $$$$\Rightarrow\mathrm{cos}\:\left(\mathrm{5x}−\frac{\pi}{\mathrm{6}}\right)=\mathrm{cos}\:\left(\frac{\pi}{\mathrm{2}}+\mathrm{17x}\right) \\ $$$$\Rightarrow\mathrm{2k}\pi\:\pm\left(\:\mathrm{5x}−\frac{\pi}{\mathrm{6}}\right)=\frac{\pi}{\mathrm{2}}+\mathrm{17x} \\ $$$$\Rightarrow\mathrm{2k}\pi\mp\frac{\pi}{\mathrm{6}}−\frac{\pi}{\mathrm{2}}=\mathrm{17x}\:\mp\:\mathrm{5x} \\ $$$$\Rightarrow\mathrm{2k}\pi−\frac{\mathrm{2}\pi}{\mathrm{3}}=\:\mathrm{17x}\:\mp\:\mathrm{5x} \\ $$$$\left(\mathrm{case}\:\mathrm{1}\right)\mathrm{2k}\pi−\frac{\mathrm{2}\pi}{\mathrm{3}}=\:\mathrm{12x} \\ $$$$\Rightarrow\mathrm{x}\:=\:\frac{\mathrm{k}\pi}{\mathrm{6}}−\frac{\pi}{\mathrm{18}} \\ $$$$\left(\mathrm{case}\:\mathrm{2}\right)\:\mathrm{2k}\pi−\frac{\mathrm{2}\pi}{\mathrm{3}}=\:\mathrm{22x} \\ $$$$\Rightarrow\mathrm{x}\:=\:\frac{\mathrm{k}\pi}{\mathrm{11}}−\frac{\pi}{\mathrm{33}}. \\ $$$$\left(\mathrm{case}\:\mathrm{3}\right)\:\mathrm{2k}\pi−\frac{\pi}{\mathrm{3}}=\mathrm{12x} \\ $$$$\Rightarrow\mathrm{x}=\frac{\mathrm{k}\pi}{\mathrm{6}}−\frac{\pi}{\mathrm{36}} \\ $$$$\left(\mathrm{case}\:\mathrm{4}\right)\:\mathrm{2k}\pi−\frac{\pi}{\mathrm{3}}=\mathrm{22x} \\ $$$$\Rightarrow\mathrm{x}=\frac{\mathrm{k}\pi}{\mathrm{11}}−\frac{\pi}{\mathrm{66}}\: \\ $$

Commented by liberty last updated on 30/Jun/21

o yes. thanks

$$\mathrm{o}\:\mathrm{yes}.\:\mathrm{thanks} \\ $$

Answered by ajfour last updated on 29/Jun/21

sin 17x+sin (5x+(π/3))=0 ⇒ 5x+(π/3)=nπ−(−1)^n (17)x ⇒ x=((nπ−(π/3))/(5+17(−1)^n ))

$$\mathrm{sin}\:\mathrm{17}{x}+\mathrm{sin}\:\left(\mathrm{5}{x}+\frac{\pi}{\mathrm{3}}\right)=\mathrm{0} \\ $$$$\Rightarrow\:\:\mathrm{5}{x}+\frac{\pi}{\mathrm{3}}={n}\pi−\left(−\mathrm{1}\right)^{{n}} \left(\mathrm{17}\right){x} \\ $$$$\Rightarrow\:{x}=\frac{{n}\pi−\frac{\pi}{\mathrm{3}}}{\mathrm{5}+\mathrm{17}\left(−\mathrm{1}\right)^{{n}} } \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com