1. ## finding the value

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

2. ## 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 $\displaystyle \sin(x)=\sin(180-x)$.

3. ## Re: finding the value

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

4. ## 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.

5. ## 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 ?

6. ## Re: finding the value

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

7. ## Re: finding the value

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