# finding the value

• Apr 19th 2012, 01:01 AM
Mhmh96
finding the value
How to find the value of this without using a calculator ?

http://im16.gulfup.com/2012-04-19/1334826074991.jpg
• Apr 19th 2012, 05:05 AM
a tutor
Re: finding the value
Using the formulae in sections 16 and 17 of this page PlanetMath you can simplify $\displaystyle \sin(10)\sin(30)\sin(50)\sin(70)$ to arrive at a value of 1/16.

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

You also need to remember that $\displaystyle \sin(x)=\sin(180-x)$.
• Apr 19th 2012, 09:05 AM
Mhmh96
Re: finding the value
Can you show me how did you apply these formulas to get these results ?
• Apr 19th 2012, 12:36 PM
a tutor
Re: finding the value
I'll get you started. I'll put in extra steps to try to be clear. I'll leave the denominator to you.

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

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

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

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

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

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

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

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

$\displaystyle =\frac{1}{16}$

The LaTeX is harder than the maths.(Whew)
• Apr 19th 2012, 01:46 PM
Mhmh96
Re: finding the value
Thank you a tutor !,now i understand how to find it ,but which formula do i have to use for tan ?
• Apr 19th 2012, 01:58 PM
a tutor
Re: finding the value
$\displaystyle \tan x =\frac{\sin x}{\cos x}$
• Apr 19th 2012, 02:02 PM
Mhmh96
Re: finding the value
So the answer is 1/3 i think ,thank you for your time .