finding the value

**Mhmh96** · April 19th 2012, 01:01 AM

How to find the value of this without using a calculator ?

**a tutor** · April 19th 2012, 05:05 AM

Using the formulae in sections 16 and 17 of this page PlanetMath you can simplify $\sin(10)\sin(30)\sin(50)\sin(70)$ to arrive at a value of 1/16.

Similarly you can get $\cos(10)\cos(30)\cos(50)\cos(70)=\frac{3}{16}$ .

You also need to remember that $\sin(x)=\sin(180-x)$ .

**Mhmh96** · April 19th 2012, 09:05 AM

Can you show me how did you apply these formulas to get these results ?

**a tutor** · April 19th 2012, 12:36 PM

I'll get you started. I'll put in extra steps to try to be clear. I'll leave the denominator to you.

$\sin(10)\sin(30)\sin(50)\sin(70)$

$=\frac{1}{2}\sin(10)\sin(50)\sin(70)$

$=\frac{1}{2}\left(\frac{1}{2}(\cos(50-10)-cos(50+10))\right) \sin(70)$

$=\frac{1}{2}\left(\frac{1}{2}(\cos(40)-cos(60))\right) \sin(70)$

$=\frac{1}{2}\left(\frac{1}{2}(\cos(40)-\frac{1}{2}\right) \sin(70)$

$=\frac{1}{4}\cos(40)\sin(70)-\frac{1}{8} \sin(70)$

$=\frac{1}{8}(\sin(70+40)+\sin(70-40))-\frac{1}{8} \sin(70)$

$=\frac{1}{8}\sin(110)+\frac{1}{16}-\frac{1}{8} \sin(70)$

$=\frac{1}{16}$

The LaTeX is harder than the maths.

**Mhmh96** · April 19th 2012, 01:46 PM

Thank you a tutor !,now i understand how to find it ,but which formula do i have to use for tan ?

**a tutor** · April 19th 2012, 01:58 PM

$\tan x =\frac{\sin x}{\cos x}$

**Mhmh96** · April 19th 2012, 02:02 PM

So the answer is 1/3 i think ,thank you for your time .

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