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May 5th, 2017, 05:18 AM   #1
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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 06:29 AM.
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May 5th, 2017, 05:32 AM   #2
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Originally Posted by ahmed786 View Post
I've got my exams after 4 days - please solve below questions; thanks.
Solve for θ
Sin 3θ - Sin 2θ - sin θ = 0
For this one, what values of theta here do you think will cause the statement to be true?
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May 5th, 2017, 06:35 AM   #3
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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?
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May 5th, 2017, 08:55 AM   #4
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For no1.

$\displaystyle \sin3x-\sin2x-\sin x=0$

$\displaystyle -\sin^3x+3\cos^2x \sin x -2 \cos x \sin x- \sin x=0$

$\displaystyle \sin x(4\cos^2x-2 \cos x-\sin^2x-1)=0$

$\displaystyle \sin x=0$ or $\displaystyle 3\cos^2x-2\cos x-\sin^2x -1=0$

$\displaystyle x=2n\pi,\pi+2n\pi$

$\displaystyle 3\cos^2x-2\cos x-1+\cos^2x-1=0$

$\displaystyle 4\cos^2x-2\cos x-2=0$

$\displaystyle (\cos x-1)(4\cos x+2)=0$

$\displaystyle \cos x=1$ 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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