Easy, won't you say?

Geometry Level 4

The value of

$cot (7.5°) + 4 cos (36°)$ ,

is expressible in the form,

$\sqrt{a} + \sqrt{b} + \sqrt{c} + \sqrt{d} + \sqrt{e}$ , where the variables are all positive integers.

Then what is the value of $a+b+c+d+e$ ?

1 solution

B.S.Bharath Sai Guhan
Aug 1, 2014

We know, $cot(7.5°) = (\frac{2 \times cos(15/2°) \times cos(15/2°)}{2 \times sin(15/2°) \times cos(15/2°)}) = \frac{1+cos(15°)}{sin(15°)}$ .

After some manipulations, we get this to be equal to $\sqrt{2} + \sqrt{3} + \sqrt{4} + \sqrt{6}= \sqrt{2} + \sqrt{3} + 2 + \sqrt{6}$

Similarly, we get $4 \times cos(36°) = 1 + \sqrt{5}$

Hence, $cot(7.5°) + 4\times cos(36°) = 1 + \sqrt{2} + \sqrt{3} + 2 + \sqrt{5} + \sqrt{6} = \sqrt{2} + \sqrt{3} + \sqrt{5} + \sqrt{6} + \sqrt{9}$

And consequently, $a+b+c+d+e = 2+3+5+6+9 = \boxed{25}$

Note that you have 6 terms in the end, as opposed to 5 stated in the problem.

The answer should be 2+3+5+6+9 = 25. I have updated it accordingly.

Calvin Lin Staff - 6 years, 10 months ago

A typo first line last term numerator the sign should be - .
I did the same way combining 1 +2 into 3= $\sqrt9$ .

Niranjan Khanderia - 4 years, 11 months ago

Calvin, Thanks!! I have edited my solution accordingly. :)

B.S.Bharath Sai Guhan - 6 years, 10 months ago