1 answer

Need help wirh all three 5. Solve the equations: (a) cos(x + 1) - cos x...

Question:

need help wirh all three
5. Solve the equations: (a) cos(x + 1) - cos x – 1= 0 (b) cos 2x – cos 6x = 0 (c) sin 2x sin x = cos x
5. Solve the equations: (a) cos(x + 1) - cos x – 1= 0 (b) cos 2x – cos 6x = 0 (c) sin 2x sin x = cos x

Answers

(a)

\cos(x+\pi)-\cos x-1=0

Solution:

\\\cos(x+\pi)-\cos x-1=0\\\cos \left(x+\pi \right)=\cos \left(x\right)\cos \left(\pi \right)-\sin \left(x\right)\sin \left(\pi \right)\\\mathrm{Simplify}\:\cos \left(x\right)\cos \left(\pi \right)-\sin \left(x\right)\sin \left(\pi \right):\quad -\cos \left(x\right)\\-\cos \left(x\right)-\cos \left(x\right)-1=0\\-2\cos \left(x\right)-1=0\\-2\cos \left(x\right)=1\\\cos \left(x\right)=-\frac{1}{2}\\\\\mathrm{General\:solutions\:for}\:\cos \left(x\right)=-\frac{1}{2}\\\\x=\frac{2\pi }{3}+2\pi n,\:x=\frac{4\pi }{3}+2\pi n

(b)

\cos 2x -\cos 6x=0

Solution:

\\\cos 2x -\cos 6x=0\\\mathrm{Use\:the\:following\:identity}:\quad \:-\cos \left(s\right)+\cos \left(t\right)=2\sin \left(\frac{s+t}{2}\right)\sin \left(\frac{s-t}{2}\right)\\\\2\sin \left(\frac{-2x+6x}{2}\right)\sin \left(\frac{2x+6x}{2}\right)=0\\\\\mathrm{Simplify}\:2\sin \left(\frac{-2x+6x}{2}\right)\sin \left(\frac{2x+6x}{2}\right):\quad 2\sin \left(2x\right)\sin \left(4x\right)\\\\

\\\mathrm{Solving\:each\:part\:separately}\\\sin \left(2x\right)=0\quad \mathrm{or}\quad \sin \left(4x\right)=0\\\\\sin \left(2x\right)=0\quad :\quad x=\pi n,\:x=\frac{\pi }{2}+\pi n\\\\\sin \left(4x\right)=0\quad :\quad x=\frac{\pi n}{2},\:x=\frac{\pi }{4}+\frac{\pi n}{2}\\\\\mathrm{Combine\:all\:the\:solutions}\\\\x=\pi n,\:x=\frac{\pi }{2}+\pi n,\:x=\frac{\pi n}{2},\:x=\frac{\pi }{4}+\frac{\pi n}{2}

(c)

\sin 2x \sin x=\cos x

Solution:

\\\sin 2x \sin x=\cos x\\\sin \left(2x\right)\sin \left(x\right)-\cos \left(x\right)=0\\\mathrm{Use\:the\:following\:identity}:\quad \sin \left(2x\right)=2\cos \left(x\right)\sin \left(x\right)\\-\cos \left(x\right)+2\cos \left(x\right)\sin \left(x\right)\sin \left(x\right)=0\\-\cos \left(x\right)+2\cos \left(x\right)\sin ^2\left(x\right)=0\\=\cos \left(x\right)\left(-1+2\sin ^2\left(x\right)\right)\\=\cos \left(x\right)\left(\sqrt{2}\sin \left(x\right)+1\right)\left(\sqrt{2}\sin \left(x\right)-1\right)=0\\

\\\mathrm{Solving\:each\:part\:separately}\\\cos \left(x\right)=0\quad \mathrm{or}\quad \sqrt{2}\sin \left(x\right)+1=0\quad \mathrm{or}\quad \sqrt{2}\sin \left(x\right)-1=0\\\\\cos \left(x\right)=0\quad :\quad x=\frac{\pi }{2}+2\pi n,\:x=\frac{3\pi }{2}+2\pi n\\\\\sqrt{2}\sin \left(x\right)+1=0\quad :\quad x=\frac{5\pi }{4}+2\pi n,\:x=\frac{7\pi }{4}+2\pi n\\\\\sqrt{2}\sin \left(x\right)-1=0\quad :\quad x=\frac{\pi }{4}+2\pi n,\:x=\frac{3\pi }{4}+2\pi n

\\\mathrm{Combine\:all\:the\:solutions}\\\\x=\frac{\pi }{2}+2\pi n,\:x=\frac{3\pi }{2}+2\pi n,\:x=\frac{5\pi }{4}+2\pi n,\:x=\frac{7\pi }{4}+2\pi n,\:x=\frac{\pi }{4}+2\pi n,\:x=\frac{3\pi }{4}+2\pi n

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