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May 5th, 2017, 04:18 AM  #1 
Newbie Joined: May 2017 From: Pakistan Posts: 7 Thanks: 0  Answer to these questions ?
I've got my exams after 4 days  please solve below questions; thanks. Solve for θ sin 3θ  Sin 2θ  sin θ = 0 Solve for x √3tan x  sec x  1 = 0 Solve: 1 + cos 2θ =  cosθ Solve sinθ2  cosθ = 0 Solve 5 sinθ cosθ = 1 Solve 3sin²θ  sinθ = 1/4 Please thanks I will be very thankful. Last edited by skipjack; May 5th, 2017 at 05:29 AM. 
May 5th, 2017, 04:32 AM  #2 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,088 Thanks: 366 Math Focus: Yet to find out.  
May 5th, 2017, 05:35 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 16,919 Thanks: 1253 
0 = sin 3θ  sin 2θ  sin θ = 2cos 2θ sin θ  sin 2θ = 2sin θ (cos 2θ  cos θ) Hence sin θ = 0 (which implies θ = k$\pi$) or 2θ = ±θ + 2k$\pi$, where k is an integer. Can you finish from there? Multiplying √3tan x  1 = sec x by (1/2)cos x gives cos(x  2$\pi$/3) = cos($\pi$/3), so x  2$\pi$/3 = ±$\pi$/3 + 2k$\pi$, where k is an integer. Can you finish from there? 1 + cos 2θ =  cosθ implies 2cos²θ =  cos θ, so cos θ = 0 or 1/2. Can you finish from there? I didn't understand your equation sinθ2  cosθ = 0. 5 sinθ cosθ = 1 implies sin 2θ = 2/5. What progress can you make from that? 3sin²θ  sinθ = 1/4 implies 3(sin θ + 1/6)(sin θ  1/2) = 0, so sin θ =  1/6 or 1/2. Can you finish from there? 
May 5th, 2017, 07:55 AM  #4 
Senior Member Joined: Sep 2013 From: Earth Posts: 827 Thanks: 36 
For no1. $\displaystyle \sin3x\sin2x\sin x=0$ $\displaystyle \sin^3x+3\cos^2x \sin x 2 \cos x \sin x \sin x=0$ $\displaystyle \sin x(4\cos^2x2 \cos x\sin^2x1)=0$ $\displaystyle \sin x=0$ or $\displaystyle 3\cos^2x2\cos x\sin^2x 1=0$ $\displaystyle x=2n\pi,\pi+2n\pi$ $\displaystyle 3\cos^2x2\cos x1+\cos^2x1=0$ $\displaystyle 4\cos^2x2\cos x2=0$ $\displaystyle (\cos x1)(4\cos x+2)=0$ $\displaystyle \cos x=1$ or $\displaystyle 4\cos x+2=0$ $\displaystyle x=2n\pi$ or $\displaystyle \cos x=\frac{1}{2}$ $\displaystyle x=\frac{2\pi}{3}+2n\pi,\frac{4\pi}{3}+2n\pi$ Conclusion, $\displaystyle x=2n\pi,\pi+2n\pi,\frac{2\pi}{3}+2n\pi,\frac{4\pi} {3}+2n\pi$ 

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