Trigo-system

May 2009
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ALGERIA
Solve : \(\displaystyle sin(x)sin(y)=\frac{\sqrt{3}}{4}\)
\(\displaystyle cos(x)cos(y)=\frac{\sqrt{3}}{4} \)
 
Dec 2009
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Solve : \(\displaystyle sin(x)sin(y)=\frac{\sqrt{3}}{4}\)
\(\displaystyle cos(x)cos(y)=\frac{\sqrt{3}}{4} \)
Dear dhiab,

Use the trignometric identities given below. Then you would be able to solve your problem.

\(\displaystyle cos(A+B)=cosAcosB-sinAsinB\)

\(\displaystyle cos(A-B)=cosAcosB+sinAsinB\)

Hope this will help you.
 
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Jul 2009
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Zürich
Solve : \(\displaystyle sin(x)sin(y)=\frac{\sqrt{3}}{4}\)
\(\displaystyle cos(x)cos(y)=\frac{\sqrt{3}}{4} \)
It follows that:
\(\displaystyle \cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)=\frac{\sqrt{3}}{4}-\frac{\sqrt{3}}{4}=0\)

Hence,

\(\displaystyle x+y=\frac{\pi}{2}+n\pi, \qquad n\in\mathbb{Z}\)

Now plug \(\displaystyle y=\frac{\pi}{2}+n\pi\) into one of the above equations to learn what additional conditions \(\displaystyle x\) and \(\displaystyle y\) have to satisfy...
 
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Jun 2009
806
275
Solve : \(\displaystyle sin(x)sin(y)=\frac{\sqrt{3}}{4}\)
\(\displaystyle cos(x)cos(y)=\frac{\sqrt{3}}{4} \)
\(\displaystyle cos(x)cos(y) + sin(x)sin(y) = \frac{\sqrt{3}}{2}]\)

\(\displaystyle cos(x-y) = \frac{\sqrt{3}}{2}\)

x - y = π/6 .......(1)

If you subtract the above two equation you will get

cos(x+y) = 0 of x+y = π/2.....(2)

From eq.1 and 2, find x and y.
 
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Grandad

MHF Hall of Honor
Dec 2008
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Hello dhiab
Solve : \(\displaystyle sin(x)sin(y)=\frac{\sqrt{3}}{4}\)
\(\displaystyle cos(x)cos(y)=\frac{\sqrt{3}}{4} \)
Using \(\displaystyle \sin x \sin y = \tfrac12\big(\cos(x-y) -\cos(x+y)\big)\):
\(\displaystyle \sin x\sin y=\frac{\sqrt{3}}{4}\)

\(\displaystyle \Rightarrow \cos(x-y)-\cos(x+y) = \frac{\sqrt3}{2}\) ... (1)
Similarly:

\(\displaystyle \cos x \cos y =\frac{\sqrt{3}}{4}\)


\(\displaystyle \Rightarrow \cos(x-y)+\cos(x+y) = \frac{\sqrt3}{2}\) ... (2)

Add (1) and (2):
\(\displaystyle \cos(x-y) =\frac{\sqrt3}{2}\)

\(\displaystyle \Rightarrow x-y = 2n\pi\pm\frac{\pi}{6}\) ... (3)

Subtract (1) and (2):
\(\displaystyle \cos(x+y)=0\)

\(\displaystyle \Rightarrow x+y = 2n\pi\pm\frac{\pi}{2}\) ...(4)

Add (3) and (4):
\(\displaystyle x=2n\pi \pm \frac{\pi}{3}\)

\(\displaystyle \Rightarrow y = 2n\pi\pm\frac{\pi}{6}\)
Grandad